Yes and no.
New drives have a large pool of extra sectors, hidden and reserved - this is in the order of tens or hundreds of thousands of sectors. When a bad sector is detected the drive automatically assigns one of those sectors as the replacement sector for the bad one, and keeps track of it in an internal mapping table so that any read/write request gets redirected.
When that spare pool is exhausted the drive starts reporting actual bad sectors, as it can no longer hide them. By this point however, the drive is far from being reliable as it has actually been failing for a while, and performance suffers as the heads keep being redirected to some other section of the platter. Bad sectors due to dust or head degradation tend to be contagious to the sectors next to a bad one, so a drive can quickly die after starting to display bad sectors.
SMART can report on the current number of reallocated sectors - this is the value to look for. Any drive with a value larger than zero here is, in my opinion, dangerous and should be replaced.
Edit: On top of this there's the Factory Defect table which is another list of hidden bad sectors detected during manufacture testing. This is usually completely hidden and can only be read with special software provided by the manufacturer. This is called the P-List (Primary defects list); the one above is the G-List (Grown defects list). Nowadays the P-List should be empty for almost all drives, but there's no easy way to tell.