I bought my PC from GE earlier this year and (touch wood) it has been fine, though I did have a similar problem.
When my PC arrived I was concerned that the soundcard upgrade was not included. I was happy to open the PC box and check that all the plugs and connectors were in place and found the soundcard had slipped out of its holding! Pushed back into place, fixed with a screw and (lucky for me) everything worked fine.
While a loose connection is a "fault" I wouldn't consider is serious enough to merit cancellation of the contract.
Personally I would open up the box and check the connections, which is quite easy to do. If you prefer not to do this then you could use a professional (or PC friend) to check the connections/ fault. However, if you go down this route it seems unlikely you'll be reimbursed for any costs you incur unless the check proves the fault is more than a loose connection.
From comments on this Forum it does seem that GE's build quality is poor and their after sales service is even worse. I wish you luck in reclaiming the cost of your power cable and that your wonderful new PC is working properly very soon.