In reply to Heathrow Director:
You were right! When I was at LATCC (Mil), we had an (almost) strict '3 a/c per console' policy. But, as you point out, these were VFR & IFR mixed, some were over the North Sea doing whatever they wanted. Some had to be thread, like a needle, through the Ambers. Sometimes 5 aircraft was boring, sometimes 1 was enough to make you cry!
In answer to the OP:
Yes, an ATCO can normally specify a limit. A Tower controller can instruct Approach to 'hold off' and a sector controller can hope that Flow Control will keep him sane. In between, most units will have some sort of policy. Some times it works, sometimes it doesn't, but I think most will agree that ATC work nowadays is a lot safer for all concerned than it was in the 60s & 70s when you just did your best with whatever was thrown at you, eh Bren?