The rules state that you make an ability check when the DM calls for it. Therefore, if 1 + your bonus is higher than the DC, the DM can decide to let you auto-succeed instead. Bear in mind that if crit fails on a natural 1 are allowed (they're an optional rule), he might still have you roll.
PHB, page 173:
An ability check tests a character’s or monster’s innate
talent and training in an effort to overcome a challenge.
The DM calls for an ability check when a character or
monster attempts an action (other than an attack) that
has a chance of failure. When the outcome is uncertain,
the dice determine the results.
As per RAW, a DM is not supposed to call for an ability check that cannot fail. Your DM might deviate from this rule, but unless he allows crit fails, it will just be a waste of time. Mind you - he might not know your bonuses, and unless he tells you the DC you're attempting, you won't know when you cannot fail due to bonuses. Therefore, you could be making unnecessary rolls without knowing it beforehand.
The same thing obviously goes the other way - if your bonus is not enough to make the check even if you roll a 20, then the DM probably doesn't think it's possible to succeed and shouldn't make you roll. Again, if natural 20s equal automatic success, you might still have to roll.