In some cases, yes. For instance:
If you have three prior misdemeanor DUI convictions within the past ten years, a new DUI can be charged as a felony.
If you have a prior theft-related conviction and served any jail time (even a brief "book & release" where you were fingerprinted and photographed, and released a couple of hours later), a new petty theft can be charged as a felony.
If you have a prior conviction for indecent exposure, Penal Code 314, any subsequent indecent exposure can be charged as a felony.
Those are just three examples that immediately came to my mind. There are probably more that I've overlooked.
Unless there are specific statutes like those, multiple misdemeanors don't just turn into a felony.
For other examples -
A prior conviction for annoying or molesting a child (Penal Code 647.6) will make a new allegation of similar conduct a felony.
If you have a prior felony DUI, a new DUI that would ordinarily be a misdemeanor would be a felony.
Aggregate thefts by an employee from their employer that add up to over $400 can be a felony, rather than a series of misdemeanor thefts.
There are other non-examples, though. For example, public intoxication [Penal Code section 647(f)] is a misdemeanor, no matter how many times you're convicted.
It depends on the specific charges and what the Penal Code allows.