Yes, it is possible to be addicted to weed.

it is actually possible to be addicted to just about anything.
However, marijuana is not physically addictive, in the sense that you will experience physical withdrawls. You will however, deal with psychological withdrawl..this is where the cravings come from.

You might try tapering down, instead of going cold turkey off it. This helps some people. For others it just leads to going off course.