You could do it if that is what you have. I'm sure a bit of stem and spacer adjustment would give you an acceptable fit.
I've never heard of it done with a modern bike. For one thing, drops and STIs are not necessarily cheaper than a set of aerobars and bar end shifters. However on older bikes things were far more interchangeable.
Personally I wouldn't do it - I'd decide if I wanted to ride a TT bike or a drop-barred bike, sell either my frame or bars and buy the appropriate ones, so I would get something which would give me a better result with less fettling. But if you, say, want to ride TTs in a few months time but use drops for now, why not give it a go?