When a promise is made but isn't kept in a timely manner, under the best of circumstances there is client disappointment and diminishment of trust. Under the worst, it can result in complete loss of trust and even the relationship. It doesn't take long to consume a stockpile of goodwill from past good performances with current weak ones.
Before making that next commitment, there is nothing wrong with pausing for a moment and considering available resources and other commitments that would be impacted by making a new one. Also consider the degree of importance of the current situation; is it Critical, Important, or simply Routine? How will the fulfillment affect the client? Promise and then schedule accordingly.
More often than not, unless a true fire is burning (or a broken mainline is flooding) there is sufficient time to be realistic, reasonable and still very responsive in the client’s eyes. By setting a practical time-frame for completion, you will be in a much better position to handle that along with whatever else the day brings your way. If possible, allow yourself more time than you believe you will actually need and then make it happen.
In most cases, your clients will not recall exactly what you did, or did not, do as promised, but they will absolutely remember if you delivered on time. In a client’s exact words: “Live up to your promises”. To that I will add: Always Deliver!