Per Article 19 "The carrier is liable for damage occasioned by delay in the carriage by air of passengers, baggage or cargo. Nevertheless, the carrier shall not be liable for damage occasioned by delay if it proves that it and its servants and agents took all measures that could reasonably be required to avoid the damage or that it was impossible for it or them to take such measures."
Where this gets a little tricky in the US is that FAA regulations (each ICAO nation independently enforces regulations related to international agreements) specifically give airlines a pass for weather related irregular operations (irrops). The trick is that airlines LIE all the time about this, such as the big blow-up around Christmas last year with Southwest Airlines where they initially claimed it was caused by weather and denied compensation, however later started offering compensation after getting called out by the Department of Transportation. The actual issue was caused by a computer systems problem that affected their crew scheduling software. So just like in the OP, airlines will lie. If you can prove their statement is a lie, you can often get just compensation.
The other challenge in the US is that the FAA doesn't take any actions on behalf of consumers in these matters, unlike the role of civil aviation authorities in Europe under the EU treaty. The FTC also rarely gets involved, even though as a consumer-protection item it falls under their purview. If an airline refuses to pony up under Article 19, you usually only have the option to sue them in the US.
So, to summarize:
1. You can be compensated for a flight delay or cancellation in the US /if/ it's an international flight under Article 19 of the Montreal Convention (MC99)
2. Compensation is not required per FAA regulations for weather-related delays/cancellations
3. Airlines often lie and claim weather when it's not weather-related, if you can catch them they'll often pony up.
4. If the airline denies your claim, your only recourse is a lawsuit, however it can be handled in small claims.