Intermittent connections over the period of months

I have been having trouble with my internet connect dropping over and over again for months.   I have tried calling the tech support thing but I always get the same solution which only provides a temporary fix.  (the solution is to unplug everything and restart the modem and router which does fix the problem but only for a couple minutes)

My set up is a modem that I got from cox connected to a linksys router which is wired to two computers.  In addition to this I have a tablet and wii that connect to the router through wireless.  

Let me add some details of my problem and maybe it can help your analysis.  

1) the disconnects seem to be related to usage.  If I am downloading stuff or streaming multiple movies the disconnects become much more frequent but if I just sit and browse internet websites everything seems fine usually although disconnects still occur.  

2)The problem is not my router set up.  To explain, I was worried my router was going bad so I bought a brand new linksys router and the problem was unchanged.  Also I took the network down and connected a single computer directly to the modem I got from cox and the problem continued.  I dont know how else to eliminate router/home network related problems.   

3) The problem is solved if you reset the modem.  you can get connectivity back but then it will drop again some random amount of time later.  

4) this has been going on for months and basicly has ruined every attempt I have made to do online gaming.  

I have come to the conclusion that it must be either the modem itself is going bad or somehow the cox network is having trouble assigning my IP or something and I keep getting disconnected for some reason.  I cannot figure out why though and it is extremely difficult to troubleshoot because no matter what you try, it will fix it for awhile, sometimes hours, then suddenly it will drop you 5 times in 5 minutes.  I have been wanting to call cox and ask if they will send me a different modem to see if that is the problem but the automatic system takes so long and you cannot explain that you have already tried a ton of things it wants you to do.