Is there so much time an investigator has to investigate a case and charge someone with something or can they take all the time they need to investigate a case, particularly Grand larceny and petty larceny in the state of New York?

Once a person is charged with the crime, the SOL stops, then you have the issue of speedy trial, unless you are out on bail, then normally speedy trial does not apply either.

If the person is not charged with the crime yet, thent he SOL applys, each state has their own SOL time frames for each type of crime