No, database errors should not be fatal and end your execution - if not your specification tells you otherwise.
As @PeeHaa say, errors should throw an exception instead of killing the script. Certain drivers might have this option as per default, but usually with the MySQL driver you'll need to set the error mode when connecting.
It is probably recommended that you allow for Exceptions to be thrown, so that you can catch the errors and perform alternative measures if the main procedure fails.
try {
$sthmt->execute();
}
catch (Exception $error) {
// Opps, the statement failed, revert or initiate error procedure
}