When do you use ability checks without a specific skill?

When, if ever, as a DM would you ask players to make an ability check without allowing them to apply a skill proficiency?

The Player’s Handbook (p. 174) says:

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. For every ability check, the DM decides which of the six abilities is relevant to the task at hand

and

Sometimes, the DM might ask for an ability check using a specific skill

The following section on using each ability lists other checks that aren’t named under specific skills. Are these tasks (e.g. Forcing open a stuck, locked, or barred door or breaking free of bonds) ineligible for gaining proficiency?