Thanks all for your input, however as I mentioned above I do verify employment, tax transcripts and credit check, etc all within a couple of hours.
Yes, I always see a red flag if if I sense they rush because they're having trouble with previous landlord [you get a sense about people in the first 2 minutes after meeting so many prospects over time]...
However, this frenzied rush from qualified A+ tenants is driven from inventory contraction in a very hot rental area, where rentals are taken down within a matter of hours (this is Manhattan fwiw)..
As Elizabeth mentioned above, because I wanted to sleep over it for '2 days,' (more of a personal thing), they took their premiums elsewhere. In the days of linkedin, instant credit check, quick employment and salary verification (even IRS transcripts is going online), I think I have to start accepting VERIFIED tenants on the spot so as to not leave premium money on the table.