Why/How a Gaussian Channel is both band-limited and Power limited?
What is usually meant by a Gaussian channel is a channel with additive (white) Gaussian noise (AWGN), which simply means that the received signal will equal the transmitted signal plus some unknown noise with a certain power and a Gaussian probability density function. A pure AWGN channel is neither band-limited nor power-limited. However, these restrictions are present in real-world systems, so it is usually assumed that the channel is a band-limited AWGN channel. Furthermore, it is normally assumed that the transmitter can transmit with some given maximum power. This latter fact will result in a bound on the achievable signal-to-noise ratio (SNR).

So, to answer your question, the finite band-width and the power limitation are realistic constraints, but they have nothing to do with the AWGN channel, which only specifies the type of noise (Gaussian) and how it influences the signal (by addition).