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With lossy encoding, what you're saying can sort of happen. MPEG audio uses a cosine transform. It basically takes the time-domain sample rate, and converts it into frequency domain. It than can analyze that frequency information, and try to find slices of a cosine wave which fit the original wave form. This Fourier transform in theory can be lossless, but it would take an infinite number of slices to completely reproduce an original analog signal.

Luckily CD audio has only 44,100 samples a second (time domain), and when converted into the frequency domain the highest frequency which can be reproduced is 22.05 kHz (Nyquist limit). So using 22k cosine slices for every second of audio would be able losslessly recreate the original PCM (within some very small margin of error). In practice that's a worse case, and one can get by with many less pieces and still have near lossless encoding.

But the problem is storing that many cosine coefficients would take more room than the original 2-byte (16-bit) samples. So in order to get the 10:1 compression which is common from MP3s, even fewer slices are used. This only gives a rough approximation of the original waveform. Fine details (like high frequency information) gets smeared together, and large rapid changes cause pre-echos or ringing.

None of this applies to truly lossless encoding schemes. They may use some perceptual estimation routine (DTS does this, Dolby's MLP along with FLAC use strictly numeric prediction) for the initial data reduction. But it goes one step further, and then checks this estimation, and looks back at the original recorded value. If there is any difference that's stored too. So as the final decode is done by the player/receiver it comes up with the estimated value, and then applies any correction to get the exact original data back. There's no loss or error.