Short term - no. I bet if you spent some time hunting the web you'd find an abundance of companies called Path, fully or as a part of their name. Right now you're too small for them to care.
If you become very successful, and you're a competitor - perhaps. But at that stage you're more likely to let your investors and their lawyers duke out the battle (or in reality settle it over dinner, since they're all basically friends anyway).
But I'd like to ask: Are you sure you want to? Path.com is undeniably getting a lot of traction and press. If they're gaining momentum, and people may confuse the two companies, the smaller company is the one that will loose the battle. Just sayin'.