The salt is usually in the db as you should have a different salt per password. The purpose of the salt is to add some complexity so that each password is in fact hashed by a slightly different algorithm (in order to defeat pre-computation based attacks.) If the attacker does not know that salt, then he does not know the full hash algorithm and cannot brute-force the password. So technically no, I don't think so. However, if the attacker can get your password db he can also get your salt db so I wouldn't rely on this by any means.

Disclaimer: I'm absolutely not a security professional.

It wouldn't, but it's a Bad Plan not to have per-user salt, so the circumstances where you'd have the passwords but not the salts that go with them would be... odd.