It would be rare for a Federal refund deposit and State refund deposit to both arrive on the same day. They come from two completely different tax agencies, each doing their own thing. Even in the odd coincidence that they do arrive on the same day, it would be separate deposits.
Only the state tax agency will know what all they are doing with your return. Some states are quicker than others; some are quite slow. And any of them can decide to examine a return more closely for various reasons. Some states have announced considerable delays this year, some by several weeks, as they try to crackdown on the large number of fraudulent returns. What state is it? We may be able to tell you if your state has announced delays.
If you efiled, you can double-check with this tool to be sure the state return was efiled successfully and accepted:
https://shop.turbotax.intuit.com/efile/efile_status_lookup.jsp
The following FAQ tells how to track a state refund. There may also be a FAQ on your state tax agency's website with into on about how long it takes to get a refund, and how long to wait before phoning them.
FAQ: How do I track my state refund?
https://ttlc.intuit.com/replies/3300211
