Yes, what matters is the size of the range, not the bounds on the range. Also keep in mind that in some cases private keys are not uniformly sampled from a range of integers; for example RSA keys, which are sampled from prime numbers.

More generally, it is common to sample a bit string and use that as a seed for a PRG, from which the private key is generated. So in that case the range is [0, 2^k] and what matters is the quality of the randomness. Unfortunately high-quality randomness is not always available, especially for embedded devices and rack-mounted "appliances."