Can the SMART status be wrong on a drive?

SMART cannot be right or wrong. It is just a report from what the disk tells it.
there are several pieces of information that come from SMART:
SMART test (pass/fail) - this is usually so serious the drive is unusable. By the time a drive fails this test, it is dead or about to be.
SMART parameters. there are over two dozen SMART values that can be checked. Drive Manufacturers use these parameters to evaluate drives that are returned to them. Some of these parameters have been found to correlate with imminent disk failure, most do not. The ones that do are:
Reallocated Sector count - this cannot change towards zero, it can only go higher. A single reallocated sector means statistically the drive will fail sooner than a normal drive.
Failed Reallocation count - the drive failed to reallocate a sector. Generally it also means the drive has started to fail and should be replaced soon.
Pending Reallocation count - Sectors which have not yet been reallocated, but have been marked to reallocate. Occasionally these sectors can "repair" themselves without reallocation, but generally, are still indicative of forthcoming failure.
Unreliable sector - the sector could not be read without retrying. This can also be the result of external factors, such as loss of power. A certify will either move this status to reallocated (meaning replace the drive), or clear it to zero, meaning the drive is probably OK.

Lastly, there are disk errors, which is a count by SoftRAID, used to inform users that the disk may be failing. Not definitive, as this can be caused by other hardware issues, but it does mean your hardware is unreliable.

Drives that are predicted to fail may work fine. Even for years. Consider a drive with one or more reallocated sectors as a person with very high Cholestrol, or very high blood pressure. They are alive, but you may not be able to purchase a life insurance policy for them!