It's not quite as black and white as Lizard made it seem, but the landlord accepting the old rent amount after the rent increase was to take effect may be considered a waiver of the increase for the months in which the landlord accepted the payments at the old rent amount. So the issue is what evidence the court has that the landlord was waiving the rent increase. In a case from over 100 years ago, which appears to still be good law in California, a commercial landlord had a month to month lease with her tenant for $65/month. The landlord in June 1908 served notice to the tenant of a rent increase to $100/month. The tenant, however, continued to pay $65/month after the rent increase. The payments were made to her agent, who accepted the payments and deposited them. The tenant claimed that the agent had orally waived the increase (which would be a problem under the rule that a contract in writing generally must be modified in writing, not by oral agreement). The agent in a statement denied having waived the increase, and the agent was dead by the time of trial, so he could not testify on the matter. The trial court found that the fact that the agent took the $65 checks and deposited them without objection was evidence of a waiver and found for the tenant on that issue. The California Supreme Court affirmed, stating:
But it sufficiently appears from the unconditional acceptance by the agent of the lesser amount of rent that the oral agreement became executed and therefore binding. It is argued that Mrs. Alden knew nothing of this purported waiver, and this is doubtless true. Yet she had clothed her son as her agent, actual or ostensible, with sufficient power to make the waiver, and she is bound by his conduct in so doing. True it is, also, that Haile, who at the trial was dead, had denied by verified answer that he had ever agreed to such a waiver. True it is that in other respects the evidence upon the matter is sharply conflicting, but, as has been said, there is sufficient to support the finding of the court in this regard.
Alden v. Mayfield, 164 Cal. 6, 9, 127 P. 45, 47 (1912). Though old, the case appears to still be good and has not been overturned. That's not surprising; the principle is still a good one in most, if not all, states. The landlord accepting the old rent amount without objection is evidence supporting a tenant's claim for waiver. Other facts might rebut that implication of waiver, so it is not the case that accepting the old amount would doom the landlord to losing the increase, but it certainly can have that result. For that reason lawyers, including myself, will caution a landlord about accepting such checks and advise the landlord to return the check and insist on the proper amount (which IMO is the vastly preferred thing to do) or at least promptly notify the tenant, preferably in writing, that the check is accepted only as partial payment of the rent and that the tenant still owes the difference. Taking the check for the old rent amount, depositing it, and saying nothing at the time simply invites a conclusion that the landlord waived the extra rent for that month. Particularly in tenant friendly California, I think a landlord in that position will often be SOL.