Not every time, only at the start of a quoted sentence. If you interrupt a quote, the second part of a sentence would not be capitalized.
"The crux of the biscuit," a wise man once told me, "is the apostrophe."
If you are reporting a sentence fragment, use the ellipsis mark (...) to indicate the missing words.
I heard Frank muttering something under his breath, "...not very nice," was all I heard.