Can I get a dry socket 6 days after my tooth was pulled?

Possible . It's possible that it's a dry socket. But, it's more likely that it's an infection or a 'bone spur'. Contact your surgeon.
Yes, it's possible, The usual time frame is 3 to 5 days after removal. If you smoke, that increases the likelihood of a dry socket. I fyou had an stitches placed that reduces the possibility as does an clotting aids used at the time of surgery.