I don't know what "too many" is, but according to mine I have 11 open and 13 closed right now, and I'm still applying whenever a good bonus comes along.
DW and I are slow churners. Every three months, we're working on a new signup bonus (currently DW's Chase Sapphire Preferred). Once we've hit the bonus, we either keep the card open for just under a year, then close it (if there's an annual fee that won't be waived for the second year) or we just stick it in the drawer (if no annual fee) and use it once a year or so. I've been doing this for years, and my credit score is 813.
Leaving them open is beneficial because it raises your average age of credit line, and it also lowers your % of credit used.