Assuming your "big signal" is, for example, some bass drums, which produce a lot of high amplitude, low frequency content, yes: when that signal is large, the diodes remain conducting and little else gets through. However, the other signals still pass during the time when the large signal is crossing through zero. This modulates the other signals, breaking them into sidebands. This is called intermodulation distortion (IMD), and is particularly unpleasant on non-harmonic signals. Music driven into clipping sounds terrible: harsh, overdriven, crunchy and farty. A relatively simple signal, like a sine wave, has no other frequencies to interfere with, so the results aren't as disastrous, and can be beneficial (hence, guitar distortion effects).
A different type of circuit, which has the limiting effect without IMD, is called a compressor (or its inverse, an expander).
Tim