This may be covered in the DMG, or it might have been covered somewhere and I haven't seen it, but
How often can a character or characters attempt something that requires an ability or skill check?
For example, bending the bars on a gate, unlocking something, or trying to figure out what runes say. If they fail can they try again and, if so, how long after? Or does it mean the task is beyond them?
Example: character tries to bend barred gate and fails. Can he/she then turn to character B and say 'Give me a hand' and try again with advantage? That's effectively 3 die rolls on that task. DCs would need to be set at 20+ or everything might become too easy! 
Bending the bars on a gate *should* be a high DC.
But to answer your question, whether or not a failed check can be repeated is up to the DM based on the context of the task. So if there's nothing preventing them from trying again, why not let them try again?
Just read this on p. 174 of PHB: "failure, which means the character or monster makes no progress toward the objective or makes progress combined with a setback determined by the DM."
Perhaps a penalty to future attempts, not giving advantage when helping with that task etc. could be the way to go. It still allows further attempts but does take the failure/s into consideration.
According to the basic conversation of the game (Page 3, Basic Rules), the DM describes the environment, the players describe what they want to do, and the DM narrates the results of the adventurers' actions. Sometimes, when the DM determines that an adventurer's action has an uncertain outcome, he or she will call for an ability check. Therefore, every single action a player states is up to you to decide whether it works, doesn't work, or might work and therefore calls for an ability check. Think about that for a moment.
So now when you have a player describe what he wants to do - bend the bars of the jail cell, for example - you have to decide whether it works, doesn't work, or might work. Let's say you decide it might work and you call for a Strength check. The player rolls and adds his modifiers and fails to hit the DC you set. So now he tries again without any substantial change to his approach. We're back to step #3 from the basic conversation of the game again: You must narrate the results of his actions by deciding whether his action works, doesn't work, or might work. Given that you've just resolved that his stated action doesn't work, does it make sense to now say it's uncertain and call for a roll? Nope.
Your quote from the PHB is applicable as a way to have the character "fail forward," that is, succeed with a cost or complication. He blew the check, but still gets the bars bent, only it has alerted the guards, for example.
Consider also using this helpful tool to help you determine how to resolve tasks: Task Resolution Tool.
Thanks. Good tips and links, too. Cheers
Happy to help!