Lights, Camera, Action Points!
In a lot of games, you have special points you can spend for things like bonuses and rerolls and the like. Used well, and it is a powerful weapon in your arsenal to control the pace and tone of your game. Used poorly and it can undermine everything you're trying to do. We'll call them cool stuff points, because the help encourage players to do cool stuff.
First, let's talk about what cool stuff points can do. As mentioned, sometimes they give bonuses. Spend a point and get an extra +2, or +1d6, or +10 to a roll, depending on your dice system. Sometimes this has to be declared before you roll, and sometimes you can do it after you roll. If you have to declare it, it makes it a lot more uncertain. Maybe you could have succeeded without the bonus. Maybe you fail anyway. But it makes it much more likely for the points to be wasted. That's not necessarily a bad thing—the chance of failure is why we have rolls, after all. But it can be frustrating, especially if the points are hard to come by. On the other hand, if you use the bonus after you roll, you're much less likely to waste it. You'd only use the bonus when you were pretty sure the bonus would bump it up to a success. This makes it a lot more predictable, but it does take some of the tension out of the decision. Either way, this can make it easier to do things that the players might otherwise have trouble doing, or even let them do things that would normally be impossible.
Sometimes cool stuff points give you rerolls. This can help make your players more likely to take risks. If they fail, they get a second shot at it. It can give some extra survivability when they can reroll saving throws or dodge rolls, and it can make it more likely for difficult rolls to succeed. It's similar to getting a bonus, though it doesn't make it possible to succeed beyond what they normally could.
Sometimes a point gives more direct survivability. In the Warhammer Fantasy RPG, Fate Points can be spent whenever someone's character would die. Instead of dying, the character suffers some other survivable fate. Maybe they're captured instead of kills, or they're left abandoned on the battlefield, but it leads to an interesting situation where they can keep going, instead of being out of the game. In Paranoia, the clones are effectively cool stuff points you use to bring your character right back into the game. In Savage Worlds, you can spend bennies to make soak rolls and cancel out incoming wounds. In games like these, the points are a direct barrier against death. So long as you have them, you're not out of the fight. When you run out, you're in dire straits and it makes the game much more tense.
In some games, you spend points to make cool stuff possible. Sometimes you have specific abilities that require one of these points in order to work, like a psionic blast or a super power. Sometimes it's more general, like being able to declare there's a rope and a chandelier for your swashbuckler to use, or that your childhood friend happens to be in the same bar you are. The characters can use their points to directly power their ability to do cool things beyond what the normal rules allow.
There is one final thing these points can do in some games. They can be used as experience points, being traded in at the end of a session. This is a terrible idea, and I will explain why.
In the first four cases, the points serve a very important purpose. They encourage the players to do things, generally the aforementioned cool stuff. Maybe it's because they know they can get a bonus or a reroll to make success more likely. Maybe it's because they know they have more chances at survival, thanks to either rerolling a bad roll or spending a point to avoid death. Maybe the point is exchanged directly for them to do cool stuff. When you let them trade them for experience, you are effectively penalizing players who use them for any other purpose. You are incentivizing them to hang back and avoid taking risks. At the end of the day, you want your players taking risks and doing cool stuff.
Think back to every fun story someone's told you about an RPG. I'll bet you that at least 90% of the time, it was a player taking a chance and having something cool happen. Or maybe something disastrous happen. But it started with players trying something out.
Encouraging them to hoard points is worse than not having the points at all, because it encourages them to avoid risk, to avoid doing anything interesting, to hang back and let other players take chances. It penalizes them for doing the cool stuff. You want them to do cool stuff, whatever that means in the context of your game. It might be fighting mobsters, it might be getting past traps, it might be political intrigue, but you want them to be active participants.
Of course, that doesn't mean just throw points into the game without thinking about how else you're using them, especially when they have multiple uses. You need to think about how many they have and how they can get them back.
You want to make sure that the different uses are all valuable enough that players want to spend them. Getting a bonus is good, but if the other use is "don't die" in a highly lethal game, your players will probably save them for the second use.
In this specific case, whether it's a probelm is going to depend a lot on how often they expect to die. In a game where players have good odds of survival, they might just keep a point in reserve, where one where every encounter ends with a player character going down is likely to see all points saved up to get them back on their feet.
Warhammer Fantasy RPG has an interesting way of dealing with this issue. I mentioned Fate Points earlier. You also have fortune points that are equal to your fate points. Fortune points get you rerolls and a few other benefits. However, spending a fortune point doesn't affect your fate points. They refresh every day. Losing a fate point, though, reduces your fortune points, and you don't get them back automatically. You have to earn them back, generally by doing big heroic things. It lets them have both types of point. Players are still encouraged to use their fortune points without needing to worry about not having fate points later on.
Speaking of giving them back, there are a few different ways to go about it. Some games have you get them back whenever you rest. Some have it based on the gaming session. When D&D 3.5 introduced action points, it had them come back when you leveled, which made them very scarce. As mentioned above, fate points in WFRPG never come back. You have to earn more through adventurning, and that happens fairly rarely in the game. In games like FATE or Savage Worlds, you can get cool stuff points for playing up your character's weakness in a way that makes the game more interesting, or by doing particularly cool and clever things.
The flow of cool stuff points will help direct the flow of action. When players are full up on cool stuff points, they'll be willing to try lots of things, to take risks, to be badasses. When they run low, they'll be more cautious. If they know they'll get their cool stuff points back the next day, they'll be more willing to spend them, where if they know it'll be a long while before they get more, they'll be more inclined to save them up.
If you want a game that's more constantly tense, then you want to use fewer points or make them harder to get back, or both. If you want a more action-packed game with the players taking lots of risks and swinging from chandeliers and the like, then give out more or make it easier to get them back.
And you don't need to have cool stuff points. They're a great tool, but it's perfectly possible to have a great game without that specific mechanic. Think about them in the context of the rest of your game's mechanics. Try them out, mix them up and see how they affect the flow of the game. Remember, there's no substitute for basic playtesting to see how everything fits together.