http://www.narconon.org/drug-information/cannabis-3000bc-1500ad.html

Here's a website describing, on some level, hemp and cannabis in England up through the 1500s. There's no doubt that it was a commodity at the time.

https://books.google.com/books?id=zXxX3e1iXfgC&dq=marijuana+in+the+1500s&source=gbs_navlinks_s

This book describes the history of marijuana as a medical agent. It shows pipes used for marijuana by Micmac Indians, who would have met with European explorers (though the Micmac tribe was in Canada). Therefore, we can at least say that Europeans at the time of Shakespeare had a lot of marijuana, and they knew that it could be smoked.

https://books.google.com/books?id=eVU_4QMn2kAC&dq=marijuana+in+the+1500s&lr=&source=gbs_navlinks_s

This book shows that a similar thing happened in 2001. They found marijuana residue in pipes found on Shakespeare's property. It also claims that people in England did smoke marijuana at the time, but that it was new. It's likely that it would be combined with tobacco.

Overall, I'd say that yes, it's very possible that Shakespeare smoked weed. In your face, Ms. Jameson.