We consider the quantization-noise amplification problem, in nonuniform sampling of bandlimited signals at Nyquist rate or higher. We first show that it is possible to avoid noise amplification in periodic nonuniform sampling, by oversampling (above Nyquist) by a factor of the period length. We prove that entropy-coded dithered quantization (ECDQ), with such periodic oversampling, can achieve the rate-distortion function (RDF) of a white Gaussian source. In the Nyquist (average) rate case, we show that while nonuniform sampling amplifies the reconstruction error, it lowers the ECDQ coding rate (though not enough to achieve the Gaussian RDF). We then show that if the ECDQ shapes the spectrum of the quantization noise via a feedback loop (which depends on the sampling pattern), then it can eliminate the noise amplification and achieve the Gaussian RDF.