Can you provide more specifics on the computer?
--Was the computer doing anything abnormal before the problem occurred? Anything at all out of the ordinary is worth mentioning.
--Were any updates to the BIOS, drivers, etc.... done before the problem occurred?
--Are you sure that the power outlet (on the wall) you are using for the monitor(s) energized?
-If the outlet isn't energized, is your GFI tripped on that outlet?
-If it is not a GFI outlet and the outlet is not linked to a GFI protector, then check your breaker box for a tripped breaker.
-If the breaker is ok, then check the outlet wiring. Sometimes wall outlets "die", if its a bad outlet then replace it with a high quality replacement.
-If you are using a surge protector/power strip/UPS system, unplug the monitor from it and instead plug the monitor straight to a known working wall outlet.
----Any luck so far?----
--Have you tried new power and data cables?
--Are all connections inside/outside the computer tight and free of dirt/debris?
--Have you checked the solder for loose connections on the plug-in(s)?
--Is there anything in the plug-in(s) on the computer that may be preventing a good connection?
--Try the Dell Diagnostic lights tool, if the computer has them. I'm almost certain that the Dimension's all have them.
1. To use the four diagnostic lights, start with the computer fully off. Make sure you are in a position to easily see the small led's on the back of the computer case.
2. Turn the computer on, record any light patterns you see while the computer works to start-up. The 4 lights will illuminate to "no light, green light, or yellow light"
3.While recording the light colors, also note the location of each color of light.
4. Post back here with your results from the diagnostic test.
----Please take your time with this. Go through everything above and let us know what you find----