I was discussing the matter with my DM. He informed me that the DMG states (alas, I don't have a copy, or I'd provide a page) that the classes are designed and balanced with two short rests as a daily norm.
To occasionally have days with only one short rest--or three short rests for that matter--is probably not game-breaking, but it does affect class balance in a small way, as some classes have more or fewer abilities that recharge during a short rest.
If the group frequently requests more long rests than is normal for the game balance, there are a few things to consider:
First, and foremost, party enjoyment is the reason we play. Unless your campaign is meant to be gritty, challenging, and generally accepting of total party KO, then it is best to be lenient and forgiving when encounters go dramatically awry.
The DM may grant some rest and reprieve, provide means of escape, fast-forward the story to another date and time, or otherwise narrate her way around the challenge. If appropriate, the DM may simply grant that the party "come up for air" and return to the scenario later, refreshed.
Alternatively, the players may set up watches, allowing some of them a short rest while others stand guard. This kind of "cautious short rest" may be an effective tool to grant necessary reprieve while inviting the players to make choices about who gets the benefits. This approach should only be considered if your players are open to making tough choices.
If it should ever be the case that the DM wants to challenge the players or that she considers her players to be taking too many rests, she could allow a short rest, only to interrupt it with a random encounter. This can be a device to communicate that short rests are not an entitlement; however, in this scenario, your players desperately need to regroup, so this response may not be welcome.
The DM may also modify the difficulty of upcoming encounters. Sometimes hitting that sweet spot of providing a good challenge while not steamrolling over players can be challenging, and DMs should be prepared to modify the encounters if their planning went wrong or the dice rolled too unfavorably.
At the end of the day, the DM and the players cooperatively choose the kind of game they want to play. I would advise that the DM only deny short rests in the event that the daily quota is exceeded and the encounter design is intended to provoke a sense of urgency.