a bike can speed wobble at any speed.
there are so many variables, that it's possible at any speed. Any speed.
I can't believe the amount of wrong in this thread.
changing parts on the bike will simply change the speed at which the wobble will happen because the harmonics change. It's not a bad idea to change parts, but there's no point changing parts when you don't understand the cause or solution to the problem.
that's like changing the battery on a clock not keeping time, but the real problem is that one of the gears is broken in the clock.
so, solution: shift weight on bike or change speed.
Ever ride a bike with speed shimmy at 15mph? Wanna bet there was something wrong with it. Bike frames just didn't happen, someone took the time to think it out, and tried to eliminate all the problems possible. Frame shimmy being one of them. Shimmy at 15mph would pretty much kill the frame.