If you have the lowest service level they offer, then it could be the issue. Call and verify what speed levels you have, there may also be a issue with COX. PC could be an issue, if there is a virus or an application always going out to the internet.
T1 is 1.5 MB full duplex (up and down at the same time), the cox depending on which level of service he has could be 10MB down and 2 MB up (half duplex). Granted it is shared media, but should be faster than your T1 assuming he has a good connection and that level of service.