no, you can grow flowers under t5s, but the few members here who do so use some pretty expensive bulbs, color corrected, at least one UV bulb in the mix....GOOD t5s aren't cheap.
heres my experience.
i started off with t5s and cheap led flood lights from ebay. i grew some weed, and it wasn't bad, but yields were shitty. i got bigger led flood lights, 300 watt ones, and it got better, but it still wasn't what i would call good yields. then i got a 400 watt hps, and everything got better, quality went up by a good bit, yield went up by more than a good bit, it almost tripled. now i flower 4 plants at a time in 10 gallon totes, in a 4x4 tent, under a 600 watt hps. best quality and yield i've gotten yet.
cobs and quantum boards give good results as well, usually a slightly lower yield, but as good or better quality, and they finish a week to ten days earlier than the same pheno under hps.
so, yeah, you can do what you're doing, but your plants will never live up to their potential with the equipment you have.
do this grow with what you have. see how it comes out. let us know. i'm sure a lot of people here will give you good advice when you decide to upgrade