Is "I don't have enough funding" the most popular phrase used as hint for rejection
It is likely an amalgam of the truth and a gentle let-down. It's a polite way to reject someone, in that it doesn't place the failure in the hands of the applicant. There's nothing you technically did wrong. Nothing you necessarily could have changed. It just didn't work out.

It's also very likely a truthful reason for not accepting a candidate. It's possible that, given a couple more graduate funding spots (getting lucky with a grant, or promised a few TA spots for their students, etc.) that they might accept an additional student, but when those don't work out,  there's nothing for it.