This is not always the case... the DoJ reminds me of a sales organization. Crimes are sales leads. Publicized leads are usually very close to being closed(solved).
That doesn't always work though - the crime may be something the public is aware of, and then the pressure to "close" the lead is even greater. There are numerous "big" crimes that the public has no awareness of though.
The Zodiac Killer and D. B. Cooper are notable exceptions. And there are probably a bunch more.
It seems statistically unavoidable that there are always going to be some big cases that go unsolved. It'll probably become less and less common - especially since it seems inevitable that nearly all DNA will eventually be traceable via genetic genealogy databases - but some crimes just won't result in any DNA or other significant evidence being left.
That doesn't always work though - the crime may be something the public is aware of, and then the pressure to "close" the lead is even greater. There are numerous "big" crimes that the public has no awareness of though.