From your description, it sounds like what you want is something closer to

my ( $x_opt, $y_opt, $z_opt );
my $status = GetOptions( x => \$x_opt,
                         y => \$y_opt,
                         z => \$z_opt,
                       );

die $USAGE if !$status ||
              !@ARGV   ||
              ( 1 < grep $_, $x_opt, $y_opt, $z_opt ) ||
              ( $x_opt and @ARGV != 4 ) ||
              ( $y_opt and @ARGV != 3 ) ||
              ( $z_opt and @ARGV != 2 );
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The last line dies if any one of these conditions holds:

1. GetOptions failed for some reason (hence $status will be false);
2. No arguments (@ARGV is empty);
3. More than one of -x, -y, and -z were specified on the command line (the grep expression will return a list of more than one item in this case);
4. -x, -y, or -z was specified with the wrong number of arguments.

Updates: Added remarks to explain the last statement. Changed warn $USAGE and exit 1 to the less flexible but simpler die $USAGE.