I guess you're right.
Talking to the people around using the same ISP, I came to know that the chances of the problem NOT being solved by the ISP are more than 99%. They just put the blame on the modem we purchased without even checking once. Moreover while searching through the Internet, this was reported by a few people on the ISP's own forum. I guess I'd rather buy a router or a switch as you say, but can you suggest something? I saw a pocket friendly one, TP-LINK TL-WR740N 150Mbps Wireless N Router, and another from the same company, TP-LINK TL-WR841N 300Mbps Wireless N Router. If I am using an internet connection with 512Kbps speed, does it really matter which one I'd buy?