Search for a tenant advocacy group, I can't think of a single instance where you would have to provide notice of intent to vacate prior to receiving notice of a large rent increase. Usually, you have at least thirty days to accept a rent increase - make sure you read your lease cover to cover. My old lease in NYC stated the landlord had to provide notice of the increase 60 days out, and intent to resign or vacate with in 30 days. You have to be provided with ample time to review new lease terms.

The threat to sue makes me think they are looking to intimidate you right off the bat. Make sure you send written communication to the management office of the important events and dates.

"Hi, this is such and such, renting a property located at "x" address from "y" management company, with offices at "z" address. The lease is currently "x" amount to be paid monthly, this date to that date.

On "x" date, I was informed a rent increase from "x" to "y" would occur in a new lease expected to begin on "x" date. This is a total increase of "x" amount. We find this increase to be unreasonable and as a result, a new lease will not be signed by the current lessee. This communication is formal notice of intent to vacate on "x" date.

Based on verbal communication with "x" management on x date, we've been led to believe the management feels we are in breach of contract. Due to lack of sufficient notice on behalf of the management company, we respectfully disagree. While we hope this can be resolved amicably, we are prepared to retain legal counsel should it become necessary."

Obviously, do some research and get your ducks in a row before you communicate, but time is of the essence. My guess is that they are hoping you don't formally respond to the rent increase in time to decline resigning. This is assuming you only recently got the notice. If you've been sitting on it for months, you may be in trouble.