Still, I'm quite sure that there is an error in your SQL query. After re-reading this complete topic, I may have spotted the error.
Please modify your code to look something like this:
$specific_value = 'Plumbing';
$query = "SELECT * FROM table_links WHERE cat = '" . mysql_real_escape_string($specific_value) . "'";
$result = mysql_query($query);
if(!$result) die("ERROR: " . mysql_error());
while($row = mysql_fetch_array($result)) {
echo 'manufacturer' . $row['manufacturer'] . '<br>';
}
As you can see, I have modified the original query so that now the queried value of cat is in quotes. Also the mysql_real_escape_string() call was added to make sure that any special characters are escaped.
If there is some kind of error in the SQL, your script should stop and show the error message.
Please, let me know if this works.