wrote

What is the cause of a radiator which is cool at the bottom, but hot at
the top? (I understand that if the reverse were true, the radiator
would require bleeding.)

This is common if the radiator has been "throttled down", either to push
heat to other radiators, or because the room has a larger rad than is really
necessary.
Check the valves at both ends. Close the handwheel valve counting the
number of turns to close (or note the setting if you have a thermostatic
valve). At the lockshield end, remove the cap (may be a small securing
screw to take out first). Using a suitable spanner on the exposed shaft
square, close the valve, again counting the number of turns required.
Now fully open both valves and run the heating for a while. If other rads
on the same circuit get hot over entire surface and this one still has
cold(ish) bottom half, then yes the chances are you've got a sludged up rad
or feed pipes. If it gets hot all over, then it's a balancing thing and the
rad has been deliberately throttled for the reasons noted above.
If you are happy with the way the system was behaving previously and the
above test reveals no sludge, fully close both valves and re-open to the
positions noted earlier. Then walk away happy. Otherwise, best to remove
rad (method has been covered frequently - google) and check the sludge
theory by flushing and catching/straining discharge. If there are serious
sludge deposits and not just a bit of black water in the final drips, then
look for an Ed Sirrett email in this group and read all of his FAQs to learn
about the maintenance of heating systems.


HTH

Phil