No, not necessarily, it sounds like only one wire is broken inside the cable which could be the ground connection, even though the cable looks OK. It sounds like you are using self powered speakers that require a separate power supply. Try unplugging the plug to the sound card, you'll probably find the hum goes.
You are sure that the speaker jack plug is in the connection. Try plugging the line into a different port and see if that cures the problem.Its the sort of stupid thing that we all do from time to time.