In large disasters, cell phone networks have historically become so bogged down that even with fully functional towers and phones, you couldn't use the network. Perhaps sometime we'll have sufficient provisioning of cell phone towers to absorb such large spikes in usage, but we're not there right now.

Crucially, however, there is no profit in getting to the point of being properly provisioned for emergencies, and no regulation pushing for it. Land lines, in contrast, are regulated to push for availability during emergencies.