Computer shuts off if bumped

Power cords rarely go bad unless they are stressed (under furniture, crimped, etc) so that's not likely (as you have found out).

Could be a loose connection between the PSU power cord fitting and the PSU transformer - which would mean a new PSU.

More likley I would suspect another loose connection on either an expansion card (video, modem, network, TV Tuner, etc.) or a loose power connection.

I could also be a grounding issue (weak ground on screws for mobo, power supply screws, or something more obscure).

I once had a network card that would cause intermittent power issues. Once I removed it I was golden.

Check all your connections - remove and re-seat them all - data and power. Reseat all your cards (Video, modem, network card, etc.), memory, power, everything. If it's not a permanent part, remove and reseat systematically until you find the culprit.

You could also remove power and data connecitons for all non-essential hardware then reconnect one at a time and test to see if one of those items has an internal problem (non-essential would include, card readers, floppy drives, dvd/cd rom/burners, etc).

Good luck

PA