DarOne, have you tried replacing the motherboard battery?  A failing battery can not only produce time related errors, but also erronenous hardware related error messages.  First step is to replace the motherboard battery then connect everything back up to the system and see how it boots up.  If the problem continues then check the back of the system to see if the diagnostic lights indicate an error code.  If the problem continues then post back here with your Dell model number and a description of any error codes you are receiving.

Dimensions 4600 Model - I now feel really silly...  The error message:  "Previous attempts at booting this system have failed at checkpoint (time or aux).  For help in resolving this problem, please note this checkpoint and contact Dell Technical Support."    I believe this occurred due to me unplugging the printer USB cable and inserting a different one without the "safely remove hardware".

 

Then, when I unplugged all hardware and powered back on, the following appeared:  "Dell 1. Analog Input.  Self Test Feature Check" followed by four boxes (red, green, blue, white).  This occurred due to me having the monitor unplugged (talk about silly!).  All is well now.  I learned a valuable lesson!

Glad to hear it is working again and thanks for posting back with an update.