Can I be a permanent resident of the United States and Canada at the same time?

The quick answer to this question is yes.

The concept of permanent residence is different in the United States than it is in Canada. The difference in definition of the term makes it possible, albeit not easy, to maintain permanent residence in the two countries. The issue has become particularly important ever since both countries adopted the approach of regular reviews of permanent resident status: every five years in Canada, every ten years in the United States.