I have a 540c that is very, very picky about how the power supply is connected. I thought for the longest time that either it or the power brick was dead, before I realized what was wrong. Like your 550c (and we are all green with envy), my 540c worked first time and then did not.

I took it apart, expecting that some re-soldering was needed. There was absolutely nothing wrong with the soldering in the area of the power connector in the machine; in fact, it would be hard to do damage to this in a 540c, so robust is the design in this respect. What I discovered was that the little round plug on the power supply itself had an issue whereby metal connectors (female) underneath the plastics had come loose, so that an electrical connection with the (male) wires in the receptacle on the PowerBook itself was difficult to establish. There is more to be said about this, relating to the plastics on the plug - for which, search further in these posts - but to cut the story short, once I figured out the way to establish a connection, the problem being literally within the little round plug, the machine has worked just fine 100% of the time.

I would initially try some wiggling, seating and reseating of the power connector while holding down the power-on key. Maybe a knock of the plug down on a table would help matters along, as the female receptacles are basically semi-circular, elongated 'pipes' meant to receive the wires that you see protruding from the machine in the power connector area. You want them right up against the holes in the plastic centre that you can see in the power connector area.

I don't quite know how to put this better, but once you take a look, you'll figure it out.