So everyone is talking about frequencies and almost nobody mentioned modulation mode. This is important because how close you have to get in frequency depends on the mode.
A signal with information is not one frequency; it has a bandwidth, or a range of frequencies that contain the information being transmitted.
The radio has filters that allow it to receive a slice of spectrum in the "pass band" which is usually slightly wider than the signal you want.
If the mode is AM, the pass band is usually the same as the desired signal, the carrier is suppressed (hopefully), and the amplitude of the RF is converted to audio amplitude to demodulate the signal. If multiple signals are present, they will all be present in the demodulated audio, like people in a room all talking at once. If the pass band is not aligned with the transmitted signal, the audio will be clipped and distorted, although with commercial AM, the signal is so wide that you'd only notice that with loud music if it's just a little off. Also, if the frequency is not aligned, carrier suppression may fail which will add a whistle to the demodulated audio. Some advanced AM receivers are able to lock onto the carrier or look for the symmetry in the signal to lock on, but most AM receivers (especially older ones) are built too cheaply to do this and just try to suppress the whistle in the demodulated audio.
If the mode is SSB, the upper or lower range of frequencies is isolated, and the opposite side is reconstructed, and the result is then handled similarly to AM. If the frequency is not aligned with the transmitted signal, the audio will be distorted badly and garbled, as the reconstructed portion will not be correct. If it is only slightly off, the frequency of the audio will be higher or lower than the original along with some distortion, and it is sometimes possible to align the frequency by ear.
If the mode is FM, the FM signal is mostly symmetrical and the demodulation method has a capture effect that uses that symmetry to lock on to the signal. (This allows FM receivers to have sloppy frequency control while still having good quality demodulation.) If the signal is within the (usually larger) pass band, the demodulated signal won't be distorted; if it is not entirely in the pass band, it will be clipped. If there are multiple signals present, only the strongest will be demodulated, although there may be buzzing as a result of the secondary signals. If multiple signals are close to the same strength, the receiver may alternately lock on each of them in turn, which may make all of them unintelligible depending on how fast it switches between them.