Again he called the same aircraft, which
sounded almost like "Bravo 511". I decided that this could not be
for me,
although I did hear no acknowledgement of the call. He made a third
call to
"Bravo 511" telling them to "Squawk 1200, contact OSU tower on
118.8."
Again I hesitated, thinking that while this might be for me, I'd
never
having been told to squawk VFR at this point. I was about to ask him
if
these calls were for me when he then called "Archer 511, are you
still with
me?", to which I replied "Affirmative, 511."

Any chance your squelch was set too high? That can cause the first
syllable or two of an incomming call to be clipped off, distorting the
sound of the message. It might sound like the controller starts talking
before keying the mic. Ask me how I know...