Having access to credit via credit or store cards and managing those lines of credit well will actually help improve your credit rating.
If you stay within your credit limit, don't carry large balances on them and make at least the minimum payments on time, every time, you will be demonstrating to lenders that so far you've been a good candidate to lend money to.
However, every time you apply for a card the company you're applying to will look at your credit record to see how you've managed your finances in the past.
Each time a company checks your credit, it leaves a footprint on your record. If five different card issuers look at your credit rating in a month, the next company may not like the fact you have applied for so much credit in a short space of time.
So, applying for credit frequently may harm your chances of getting another card or loan, even though you're managing your cards responsibly.
If you've collected a number of cards you rarely use I would suggest getting an up-to-date credit report from a company such as Experian or Equifax and seeing exactly how many credit agreements you have in place. You may be surprised how much credit you have access to.
Check you have now qualified for and received whatever benefits you took the cards out for in the first place, pay off whatever balance is left on them and close the accounts. By all means keep a few cards for convenience or, if you use them regularly, to collect reward points but close any cards that don't see the light of day. Keep managing your remaining cards sensibly and you should have a first-class credit rating.
One last thing, being turned down for credit doesn't always mean you have a poor credit rating. Lenders make money by charging you interest so if you always pay off your balance each month, some credit card companies may feel you're just not going to be profitable for them.