Now this is a bit tricky since most of the time its not your modem that has an issue its you ISP but here are some things that can happen if you have a defective modem .
1 - Intermittent connection ( meaning that you lose connectivity very frequently )
2 - If your device has issue getting an Ip ( hardwired or wireless ) or getting unidentified network on vista or 7
3 - Slow connectivity can happen but keep in mind that for this issue is best to contact your isp so they can make everything is right on their end
4 - Loss of packets
5 - Modem is atleast 3 years old and is having any of the before mentions symptoms ( if not defective and upgrade wouldn't hurt )
6 - Red power light or no power at all
Hope that answers any questions .