I got some 1 hand trainning from Amren and I pickpocketed the gold back and got the 5 points to 1 hand skill and 4 points to my pickpocket skill. I'm guessing that because I was still getting the message that my 1 hand skill had gone up it got confused and gave me the points to my pickpocket skill as well, because my pickpocket skill also went up a point when I took the gold back.

It is perfectly possible that your Pickpocket went up by four points. Pickpocket levels faster if the goods you pickpocket are worth more, and pickpocketing money is a good way to train. How much money did you pickpocket, and what level is your Pickpocket now?

My pickpocket skill was 60 before and 65 after, I gained 1 point when I pickpocketed the gold back and got 4 points after that. I gained the point after taking the 1600 gold back, but then got the extra pickpocket points.

Sounds normal. Pickpocket levels very quickly.

The Pickpocket probably leveled normally, but you may have done it so quickly that the leveling messages got 'lost' in lag maybe?

But would I get 4 points one after the other for pickpocketing the gold back? This didn't happen when I did the same trick with Faendal. The only difference is that I gained a point in pickpocket with Amren and didn't with Faendal.