add to existing value in mysql column using CONCAT function?
it should be
UPDATE ptb_messages SET subject = CONCAT( subject, 'newvalue')WHERE ...
in PHP
$sql = mysql_query("UPDATE ptb_messages SET subject = CONCAT(subject, 'newvalue') WHERE id='".$message_id."'");
As a sidenote, the query is vulnerable with SQL Injection
if the value(s) of the variables came from the outside. Please take a look at the article below to learn how to prevent from it. By using PreparedStatements
you can get rid of using single quotes around values.