All of my professional jobs have been contingent on background checks and validating (to some degree) the things I put in the record. If I say I have a degree, they call to verify. They called to verify work history, although not being able to reach previous employers wasn't a deal breaker. I don't think just claim you have a degree when you don't works.

If you have a degree from a 'good school', that gets you some credibility by itself, but mostly a 4 year degree says 'this person can commit to doing difficult things without an immediate payoff for around 4 years' which is a valuable thing for employers.