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.

The 20mm Phalanx CIWS has proven itself in its land based version, which has been used to defend against rocket, mortar, and missile attacks on bases. The anti missile missiles on carriers such as the RIM-7 and RIM-116 have not been tested in actual combat though. Carriers usually carry a bunch of these, and are usually surrounded by other ships armed with them as well