Understudies and standbys aren’t usually rehearsed in until after opening. When one must go on out of town or in previews, it is usually a big deal (a la D’Amboise in Sweet Charity) and shows are often cancelled until the cast member gets better or the u/s s/b is prepared to go on.
Shows cancel previews/delay opening all the time. Often, they just want to make sure what they are presenting is worthy of a paying audience. Sometimes, first previews go on when they should have been delayed and it leads to bad press - Women On the Verge was a huge case of this, although delaying the previews any longer may have caused even greater press...
One of the greatest productions I’ve ever seen, Three Tall Women, cancelled the first few previews, leading to murmuring on here about trouble. I attended the first preview, and it remains one of the tightest, most perfect evenings I’ve spent at the theatre.
"Sticks and stones, sister. Here, have a Valium." - Patti LuPone, a Memoir