Historically, I've found Marriott to be reasonable about weather-related delays. As someone who used to connect via ORD to points northeast on a regular basis, I've had to make that phone call a couple of times - sometimes at 9 or 10PM on the scheduled night of arrival - and Marriott never failed to adjust my arrival date or cancel the stay at no charge.

No guarantees - they certainly aren't legally required to do this - but I do believe it's a fairly common courtesy, especially on something easily verifiable like an ORD blizzard or a CUN hurricane.