The US Navy has some of the best missile defense systems in the world right now. An Arleigh Burke Flight IIA destroyer has a 96-cell vertical launch system for missiles. Each cell could hold an SM-2ER surface-to-air missile with a ~100 mile range, an SM-6 with a ~150 mile range, an SM-3 with ~600 mile range for ballistic missiles, or 4 ESSM missiles with a ~30 mile range. On top of that you have a Phalanx Close-in Weapon System (CIWS) which is a 20-mm autocannon designed to shoot down incoming missiles up to 1 mile away. On top of that, the US Navy has a networking system that allows targeting missiles using remote sensors. So a E-2D Hawkeye early warning aircraft could detect a low-flying cruise missile traveling below the radar horizon of the destroyer, and then even though the destroyer radar can't pick it up it could launch a missile and have the Hawkeye's radar guide it to the target.

The exact numbers would depend a lot on the type of missile attacking, the angles they are coming in at, whether there are external sensors in the area, weather, etc. But a destroyer escort is more than capable of intercepting an incoming missile. In perfect conditions, where you detect and are tracking the missiles at extreme range, a single destroyer could likely take out dozens of cruise missiles. Scoring a hit on a carrier would require some combination of: advanced missiles (fast, stealthy, maneuverable, etc), multiple missiles, element of surprise, attacking from multiple directions at once, and jamming or otherwise degrading sensors.