Does anyone know whether if a site is already ranking highly for a number of keywords and then you try to get it to rank highly for another one that can hurt your position for the keywords you’re already dominating? For example, if I have a website ranking very highly for the three terms A, B, and C right now, and I start a major new backlink campaign to rank for the term D, could that potentially hurt my SERP positions for A, B, and C?
No it wont, every backlinks in which your link is placed alongwith your keyword is independent of your other backlinks. So you're linkbuilding targeting another keyword won't affect your placement for the first keyword.
There is such a thing as spreading yourself too thin. Just make sure you don't make too many pages that don't convert.
Thanks for the responses, but does anyone have any actual experiences to share that go beyond opinion?
No, this will not. But please make sure that you also do some work for previously achieved rankings. Because to maintain rankings some SEO work is always required.