Yup - that would work as a test. What you want to do is to isolate the problem, so if you can test that cable with a different device, that would be the ticket.
Just an aside....I have found that even my authorized Apple USB cable that I use normally for my iPad, is not always being recognized by my iPhone since I downloaded iOS 7. If I remove it and reconnect it, after one or two tries, it does recognize it. I know the cable is fine, because as soon as I get done with the phone, I plug it into my iPad (running iOS 5 - it's an iPad 1....) and it, without fail, connects.
So, I think with iOS 7, the connection logic is a bit more sensitive and sometimes is just not making the connection when it should. I am certain that it is iOS7 because the issue very clearly started occurring after I upgraded. It's a bit annoying, but eventually it always works, so just a bit of a nuisance. Just wanted to share that with you in case a couple or three tries might get it to connect.... (this also is happening with my Bose speaker that I dock my iPhone on to play music during the day. Again, I am certain it started happening after iOS 7, so the two seem to be connected to me....)
Cheers,
GB