Can someone who filed their taxes after you get their taxes before you?

Sure.  Happens all the time.  And federal and state come separately at different times.  So you can get one before the other.

Did you efile?  Was it Accepted?

Did you go all the way through the File tab, through the 3 steps and hit the big orange Transmit Returns Now button?

If you used the Online version you can check your efile status here     

https://turbotax.intuit.com/tax-tools/efile-status-lookup/   

When you efile you get back 2 emails.  The first email only confirms the transmission.  The second email says if the IRS (or state) Accepted or Rejected your efile.  And check your spam folder for them.

When you log into your account you should also see the status and if it was Accepted or Rejected, Started, Printed, Ready to Mail, etc.  What does it say?

If your return was Accepted then check IRS Where's my refund

https://www.irs.gov/refunds/

When will you get your refund?

https://ttlc.intuit.com/community/refund-status/help/when-will-i-get-my-irs-tax-refund/01/25589

IRS - Questions about Refunds

https://www.irs.gov/refunds/tax-season-refund-frequently-asked-questions