Have you tried just having these plugged in, modem, wireless keyboard and mouse, to see if the problem is still there.
If your PC is a laptop there could be an issue with the power your getting from the USB's, but if you have a desktop PC I would not expect any power issues with the USB's, I would have suggested changing batteries and updating the drivers, but you have already done this.
If I were you I would take it back to the shop and get them to exchange it for another one. I have a logitech keyboard/mouse which I have always found to work perfectly, however they are not cordless, but I could expect that Logitech's cordless keyboards/mouse would work in the same fashion, just a suggestion of another brand for you to consider, thats if your present mouse/keyboard are not Logitech. I would just exchange your present mouse/keyboard for now.