Your refunds are coming from two completely separate sources, the U.S. Treasury, and the state treasury of whatever state you filed in. So no telling if they will arrive at the same time.
A lot of companys will for-go their limitation on funding as long as the funding is from a Tax Revenue place. You can also call your company to let them know that you are expecting direct deposits and they can make a note on your account.
Does IRS get my direct deposit from my return if I owe IRS and a direct deposit from the state of Michigan
If you owe it is not a deposit but a Debit. You can have Direct Debit set up for IRS. Don't know about your state.
Check the printout or PDF of your return; look for the state cover sheet with the Turbotax logo. If you owed tax, it will show the payment information and how/when you decided to pay. Read the state payment instructions carefully since most states cannot be paid from within the TT system and requires additional steps.