There are thousands of banks, so no one can predict a specific time when they deposit. However, where did you see that today is your deposit date? Are you referring to the estimate in TurboTax, or were you given a deposit date at the IRS "Where's My Refund" tool?
https://www.irs.gov/wheres-my-refund
Don't go by any estimate that TurboTax provides; that's just the most common scenario. Go only by the IRS tool. The IRS also says to allow up to 5 days to show in the bank account due to variations in bank processing procedures and times.