I should have mentioned that the thumping sound corresponds with the
speed of the car - ie approximately the same speed as the wheel
revolution.   The tyres have been checked, and look OK.
John

Still sounds like a tire.  Rotate the rears to the front and the fronts to
the rear and see if it relocates or stops.  Could be a separating carcass
and tread, and in the early stages you might not be able to see it on visual
inspection.

Davehd

This sure sounds like a bad tire.
Mount the spare on one side, then the other.

could also be a half shaft, how many miles on the car?

My 1987 190E has developed a thumping noise from the rear end.
Sometimes it is there, and sometimes not - which makes it hard to
identify.
I took it in to the service station, and after searching around the
differential, could find no problem.   They thought it might have
something to do with the drive shaft, but couldn't see anything wrong.
The thumping noise stopped for a few weeks, but has come back.   It
only happens when I accelerate - stays for a while, then stops for a
while.
The car is going in for another service this week, and I hope they'll
find the cause.
Any advice?
John