How much "power" do ATCs really have?
As an active area radar controller, I completely agree with @rbp answer.
To better explain relationship between power and responsibility, I will try to describe what a controller would usually do: If I need an aircraft, for example, to reduce speed...
I know approx. what would be the minimum convenient speed for a particular type to ask, so I'd take this into account prior asking. But if a pilot tells me that he's unable to comply with the request, I'll find another solution i.e. radar vectoring to loose few miles or in extreme situation, level change. In short, a controller will never force a pilot to action if pilot answers that he can't do that. A controller must plan in advance that pilot may not be able to fulfil his request and have alternative solution. If you, as a controller, put yourself in a situation where you have no alternative, that is usually too late.
There are usually only few things where I can't make exception:


Minimum separation between aircraft
Danger/Prohibited zone
Terrain


Even then, in extreme case, I've never met: Imagine a situation where you have an aircraft which can't change level due to traffic above and below and has a crossing traffic. Then only remaining action I have available is radar vectoring, but what if the pilot advises me that his unable to turn due to weather, (the described situation is usually the controller fault because he should have predicted that earlier). Then controller probably would give the pilot traffic information (tell the pilot what to expect) and advise the minimum distance but it will not force the aircraft to CB.