Yes, if you have several separate email addresses ie firstname.lastname@example.org, email@example.com, firstname.lastname@example.org (one that you use on Gmail only, another that you use on Outlook only, and the last that you use on Office365 only). You would have 2 or 3 different services running separately and that's fine (depending on what services you want at the same time).
No, if you're only using a single email address (a Gmail or Google Apps one), and no if you're importing your Gmail into your Outlook, or Gmail into Office365, etc. This would unfortunately create duplicates. You'll need to choose which of the 3 services you prefer and stay on that platform only.
IF, you'd like to switch from Outlook to Gmail, simply follow the steps HERE and if you'd like to switch from Gmail to Outlook, simply download the outlook version and the switch'll happen automatically.