You know what's horrible when you think about your comment? Those places offer better starting pay and free food! Our starting pay for part-timers is a disgrace for the company.
I've known it to be possible, and even if UPS knew about it there was little they could do. Now if Fedex knew about it they could fire you for a conflict of interest since there is no union to protect you.
The OP never said he wanted to leave UPS--he is looking for a 2nd job and wanted to know if there would be a conflict of interest if his 2nd job was at FedEx.
I would guess when you put down UPS as your current employer you will find out if it is indeed possible.
If you work anywhere ( whether it be another company or starting your own business) as hard, as fast, as committed, etc as you do at UPS; you will be successful.