Does cancling a credit card do damage to a persons credit & Fico score?
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According to FICO closing an account does not raise scores and in fact could lower a score, in most cases it will do nothing.
As you stated the banks are "jumpy" about opening new accounts, many people have found that even with great account history, some of their cards have raised the APR and lowered the credit limits.
Even once an account is closed it is still reported on your credit report and it’s history is factored in your score.
It they turned you down for too many accounts, then that is just the reason, nothing to do with your creditworthiness, they just feel you could get yourself into deep debt.
So don’t close any accounts as it will only lower your debt utilization,
Read the source I have referenced, it is from FICO
Hope this helps answer your question
Canceling a credit card does decrease your credit score a little but I would not worry about it. In general, the newer the card, the less of an impact that it will have on on your score when it is canceled