The statute of limitations is an affirmative defense when a lawsuit has been filed or is threatened. The statute of limitations has no application outside the context of a pending or threatened lawsuit. Failure to promptly bill for services actually performed is not a defense to payment and does not implicate the statute of limitations defense.
In California most written contracts have a 4 year statute of limitations while unwritten (oral) contracts have a 2 year statute of limitations.
I agree with my colleague. Billing a client late for services rendered is poor business practice, but it doesn't trigger any statutes of limitation any kind, since statutes of limitation are defenses that can be raised by a defendant/debtor to restrict the plaintiff/credito's ability to file a lawsuit.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.