Networks

Wait, you are listing 2 different networks here.
Your machines are in the 192.168.1.0/24 network, and they can't connect to one another, but they can hit the 192.168.0.1/24 network? That doesn't make a lot of sense.
All machines on the same Class C need the same subnet mask and IP address in the same Class C.
So, say MACHINE1 has an IP of 192.168.1.2 with a subnet mask of 255.255.255.0, and MACHINE2 has an IP of 192.168.1.3 with the same subnet mask.
Both machines are plugged into the same hub or switch.
The network cards on both machines work, and they get link lights.
Then, the local IP network routing has to work.
Please verify all of this and post a reply.

Before all, make sure the source of the problem is not from the switch/hub.

Secondly, your PCs should not have duplicated IP addresses.

Do arp -a to see what physical addresses are mapped to the IPs.

Extra to Answer 1, the default gateway for those computers should be 192.168.1.255

Thanx for ur replies Guys.

But the solution was really strange, probably most of us may have never encountered the same.

The Problem was the motherboard, (VIA with onloard LAN card ), all the LAN cards had the same MAC Address !!!!!!!

The Machine that I was able to ping had an external LAN Card.