I paid the archery trainer somewhere around 4500 gold for a skill level, and then I attempted to pickpocket the gold. Even though my pickpocket is at 98 and I had on about 40+ pickpocket gear, the chance of success was 0. Is there a certain amount of gold that becomes impossible to steal?

If I were a typical Skyrim NPC (meaning, all of them are poor!) and had 4500 gold on my person, I'd be keeping them VERY close until I could deposit it in the 100% PC safe banks in the world. They are 100% safe, so safe in fact you didn't even knew they exited but where else do the merchants hide the gold they use to pay you?

Your question kinda has the answer you look for at the same time. 4500 gold is probably already past the point where you cannot pickpocket at all anymore.

I've seen this on a number of vendors. There must be a flag somewhere that says Gold in Unpickable for certain characters. No matter how good you are, the money is 100% safe.

The chance may be 0%, but if you use ice form on them (assuming they're not a vendor, as you'll be charged for assault) and wait for the second they thaw to go in for the grab, you'll get away with stealing anything off of them, including the thousands in gold. So I'd recommend investing in non-vendor trainers if you want to save your gold.

Video example:
http://www.youtube.com/watch?v=N6XgT3qgCc4

I believe that beyond 2000 or 3000 (or so, I forgot) something makes you can't pickpocket the money anymore.

probably a bug, though

I've read that you cannot pickpocket more than 2,500 gold - at that point detection becomes guaranteed. Vendors keep their stock (and gold) for trading in a hidden container in the same interior cell. usually high above the ceiling or floor in what is essentially null-space and unreachable without noclipping

This is good to know

It was the guy in the Thieves Guild, so he didn't get mad, but he didn't let me keep the gold either. I doubt if I could ice form him without aggroing the whole room.

Oh well, it IS good to know.

Ice form isn't considered aggression, fyi. :-p

It is to some people actually. Arcadia, for example, will somehow report you for assault the second she's frozen. Farska or whoever, from the companions, won't get mad though.

Raestloz is right; there's a value where you're not allowed to pickpocket successfully anymore. This applies to both lump sums of money and the value assigned to individual items. I don't know what those values are, however, but the chance is based on the items weight, value, and your skills/perks.