Trying to understand cash out refi

Suppose I buy property with cash (to get a better deal, quick close etc). Can I refi with a mortgage and get the same rate down the road?

Let me explain. Say price of home is 500K and investor mortgage rate is 4.5%. I pay full cash and get the house. 2 months later can I take a mortgage and get 400K back (20% of 500K is downpayment) and still get the 4.5%. In all of this, I am assuming price and interest rate hasn't changed for 2 months. Is there a time or financing limit to the above.