Square brackets are for matching individual characters. The regex you wrote will reject (because of ^) any word that has any of 'm', 'a', 's', 't', 'e', 'r', '|' .. and so on.

If you meant a valid database name except the names given in your regex you could have written

if($name =~ /^[a-z0-9_]$/i && $name !~ /^(?:master|model|dbccdb|sybsec
+urity|sybsystemdb|sybsystemprocs|tempdb|DBA)$/)
{
  print "allowed\n"
}
else
{
  print "not allowed\n"
}
[download]

Update: This can also be written in a single regex using negative look-ahead assertion as follows

if($name =~ /^(?!(master|model|dbccdb|sybsecurity|sybsystemdb|sybsyste
+mprocs|tempdb|DBA)$)(^[a-z0-9_]+$)/i)
{
  print "allowed\n"
}
else
{
  print "not allowed\n"
}
[download]