The most common and most likely answer to this crossword clue is the 14 letter word OVERSUBSCRIBED
I've seen this crossword clue in The Times
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More clues leading to the result OVERSUBSCRIBED
Buddy explains!

The straight part
We know that "having too many clients" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case having too many clients = oversubscribed.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does OVERSUBSCRIBED mean?
OVERSUBSCRIBED verb- To subscribe to an extent that is greater than the availability
- To use the oversubscription technique in multithreading.
- To use the oversubscription technique in a computer network.
- Having insufficient capacity to meet the demand of those who are interested.