Is there any known case where ZFC system is known to be consistent with a statement, but is also consistent with the negation of the statement? Or vice versa.

Also, when we say ZFC is consistent with something, are we saying that one particular model of ZFC allows something as being consistent?

Edit: Phrasing the question this way, I see I was somehow mixed up. Answers to two questions are obvious... For some checks, can anyone answer the second one? - though I think the answer is obviously.. (as some other models can behave differently)