Today I changed my skin five times and after that it wouldn't let me change my skin again! I want to pick a different skin because the skin I have currently doesn't look that good. Did I do something wrong or is it Minecraft?
There are limits for contacting the Mojang servers for same thing over and over. This apply for basic getting of info about player (getting UUID, nick, etc) and also for skins, capes and other info.
Be aware that if you (or the server you play on) ask too often for your new skin, the next connections will be refused by Mojang servers for some time.
If you changed it more times in small time window, Mojang servers could consider you as attacker or spammer. Waiting some time should fix this problem, try it the next day(after 24 hours) and you should be good.
If there are more issues after that time, then you can always try to contact Mojang Support.