I am not familiar with the dimensions of that case. However, it's never a bad idea to open it up, measure how much clearance you have, then check out the tech specs of the card you're considering. It will tell you how long it is.
If it's a gateway/dell/hp, you probably also want to check the PSU to make sure you have enough juice/connectors to support the card.
Is this your machine? It looks like it has a flipped ATX board so the bottom slow is actually the PCI-E x16, is that correct? You would just want to make sure the card length is ok. How much space is there from the slot mount to the first obstructions toward the front of the case? I think the 6950 is 11.3" long.