I think the time you are most likely to come up against the tyre limiting speed is on landing in a non-standard configuration e.g reduced or no flaps.
I fly an RJ, and at 45t TOW, max rotate is 142kt.
Tyre limitng speeds are 190kts nosewheel and 210kts(?) for main wheels.
These are way in excess of any take off speed, but tie in nicely with the speeds that you will be reaching if you have to do a max landing wt flapless approach.
In a Canberra jet I believe that rotate was below Vmca and so for a short while after take off it would be disasterous to have an engine failure, especially if you were the observer without an ejector seat.
I met someone who had flown one in the RAF and asked why they just didn't take off faster. "Tyre limiting speeds" was the answer
