Yes having too many credit cards open can hurt your credit score because it makes it seem like you're not responsible with credit and will have a hard time of paying off the credit card balances.
Also applying for credit cards puts a hard inquiry on your credit report which lowers your score by some points at least for awhile.
Eventually it no longer affects your credit score but every time you apply for credit cards or loans it lowers your credit score due to the hard inquiry.
Having six or more credit cards is certainly way too many credit cards and can really hurt your credit.
It's best to only open a few credit cards and use the credit cards responsibly which helps to build your credit.