Consumers routinely call to file a complaint about denied homeowners insurance claims in which the problem is that they just discovered a long term problem. These tend to be things like mold, rot, mildew or deterioration.
The problem for these homeowners is that Insurance is designed to cover sudden and accidental damage, but not wear and tear for home care and maintenance that any homeowner is expected to address on their own. Some policies may allow coverage for damage caused within a matter of weeks -- and they may mean that plural literally, as in just two weeks -- to be considered sudden and accidental.
Generally, an insurer will inspect the damage to determine if it could have been caused suddenly or over a period of time. An example is mold damage that resulted from a leaking hot water heater, or a refrigerator water line that's been leaking, slowly rotting the kitchen floor joists.
The upshot is this: the longer an underlying problem has existed, the harder it is to successfully claim that the damage was sudden.
Update (9/25/2012): Here's a classic case of this sort of problem.