Practically speaking, if one is interested in a frequency band well-separated from the line noise (say, for example, the 10-20 Hz band, with 60 Hz line noise), would it be advisable to notch out this line noise before bandpass filtering to the desired frequency range? It seems that, given of course an acceptable filter rolloff, it would be better not to perform an additional filtering step if it were not necessary; however, I'm wondering whether this may lead to some unforeseen problem.
A general rule of thumb is to filter as little as necessary, because every filter distorts your desired signal, even if just a little bit. Also: a notch filter is advised in a situation where narrow-band interference is present within your signal's band. The notch filter will reject the interference while having minimal effect (ideally) in your signal.
In the problem you describe, my preferred approach would be to go with a low-pass filter only, and increase its order to achieve sufficient rejection of the 60 Hz noise.