I'd allow this, but only with an appropriate aspect, and only for stunts that can't be represented as a numerical bonus. For example, a character who is "Crazy Prepared", might invoke that to be allowed to use sleight-of-hand, ala the pickpocket stunt - for one specific feat of pickpocketing that she'd practiced beforehand. If she wanted to use prestidigitation a second time - even in the same scene - she'd need another fate point.
If you're considering wider access to stunts, here are a few things to keep in mind:
1: Don't allow skill swap stunts. You shouldn't be able to spend a fate point to make a jump-kick using your athletics skill of 5, when your normal fists skill is 2 or less.
2: Be careful of stunts with large bonuses, like the "Listening" stunt Harry uses. That stunt grants a (if I remember right) +4, at a penalty to other forms of perception - in the case where someone wants to buy that stunt with a fate point, I'd tell the player up front, "Yes, you can do this, but I'm going to treat it as spending a fate point to declare a 'listening very intently' aspect, invoking that aspect for a +2, accepting a compel against that aspect, and using the fate point from the compel for another +2. In other words, you can have your +4 bonus, but you will be paying for it by immediately not noticing something else important."
3: Be very very careful of any temporary stunt purchasing that's going to last for a scene instead of a single action; most stunts are equivalent to spending a fate point each time the stunt's conditions come up; if I have a choice between spending a fate point to get +2 to this roll, and spending a fate point to get +2 to this roll and all similar rolls in the scene... that second option is pretty clearly out of line with the base game rules.