In your organization there will come a time when a department has an impossible request. The deadline is too close, the scope of work too great, whatever the reason, you know it cannot be done in time. Things can always get tense when the answer is a flat "no," but as a leader it's your job to find the consolation.
Your team may not be able to meet another's expectation, but is there something else, a smaller but just as significant task you can accomplish? Is there something the department has been wanting for a long time that has a low barrier to completion? Something that can act as a peace offering?
It's not usually necessary you appease a fellow department when the expectation is unreasonable, and if you simply don't have the manpower there is a logical excuse. But going the extra mile and finding that olive branch to extend can go a long way, especially when the day comes and it's you needing help.