I think "thuh" and "thee" make it easier to transition into a consonant and vowel sound, respectively. I and everyone I know just does it automatically. "Thee bottle" sounds ridiculous, and "Thuh electric toothbrush" sounds clumsy. I think most people don't even realize that they make a distinction. It just comes naturally. After all, it isn't typically memorized since it doesn't affect writing (since both pronunciations are written the same way)

But like Tim said, it's the sound, not the letter, that determines which one you should use, so it's actually "Thuh university," since university starts with a consonant sound. For the same reason, it would be "Thee hour."

Now if only I could figure out why sometimes "a" is pronounced "uh" and sometimes "ay." There doesn't seem to be any reason for it