leapingfrog I am in a similar situation to you, and the problem I am now facing is that the university where i am accepting my SSHRC award is refusing to fill out the bottom and fax Form 1 Part 2 to SSHRC before they have confirmation that I have finished my Masters degree. It's nonsense because the bottom of this form asks the uni to confirm that the student WILL be unconditionally accepted and enrolled, but my university (and I heard a similar thing from a McGill doctoral student who took up his award two years ago) takes this to mean IS accepted and enrolled, and the university won't proceed with official enrolment until I send them confirmation that I have finished everything in my MA program. So yes, you can have some slight overlap of degree programs, but you might face a block somewhere along the line and not receive any money until your university has confirmation that you have finished your thesis. I have no idea how this works for students not attending a Canadian university.
Unfortunately all this means that my payment will be delayed now.
And just a friendly reminder to the others on this forum, if you are taking up your award at a Canadian university and have not yet filled out Form 1 Part 2 and sent it to your university to fax to SSHRC -- DO IT NOW. The form has to be received by SSHRC a month before the start date of your award, otherwise your payment will be delayed.