Can i jail break an iphone 4 that has been factory unlocked to use for tmobile?

I have a 16 gb i phone 4 it is unlocked

If it is factory unlocked then why do you want to jailbreak it for T-Mobile? You don't need to, that's what unlocked means…

You should know, though, that even though it will work on T-Mobile, you won't be able to use their 3G, because T-Mobile uses a frequency that the iPhone doesn't support.

Yes, unlocked iPhones can be jailbroken. Use evasi0n for 6.2.1 firmware. Look it up, it's pretty easy to do yourself.

Jailbreak and unlock are not the same thing. Unlock means it can be used for any carrier, while jailbreak allows you to customize it more.