Good topic for discussion - I’m sure lots of people wonder about this.
I’m not 100% sure what you mean when you say “Nightcrawler”, but I suspect you might be referring to “Canadian Nightcrawlers” (aka Dew Worms - Lumbricus terrestris). Assuming this is the worm you are talking about, unfortunately they do not make good composting worms, nor are they easy to breed in captivity. If you have open-bottomed bins sitting on soil outside, there is a good chance that Dew Worms (and other soil dwellers) will venture in to the lower reaches of your bin, feeding on some of the organic matter while they are there. But these worms just aren’t suited for the crowded, warm conditions in rich organic matter that Red Worms love so much. They might survive for awhile, but you certainly won’t get a breeding population of them.

If on the other hand we are talking about ‘European Nightcrawlers‘ (Eisenia hortensis), then the answer is yes - Euros and Reds can be combined in the same bin. Some actually like having both in the same system since the Euros tend to go down where the moisture content is higher, while the Reds remain up near the top - thus ensuring that the entire contents are getting vermi-processed.

Red Worms do tend to be a more prolific and active worm however, so mixing the two might not always be the best idea if you want your Euros to really thrive. Also, it might be a bit of a pain to separate the two species later on if you want to sell them etc.