So do you want a entry like:

dn=machinename1
name=value

to yield:

$hash{"machinename1"}{"name"}="value";

or

$hash{"machinename1"}{"name=value"}=1;

in case 1 you should save the different values for name in an array. something like this should work

foreach(@files){
    my $dn = "";
    open FILE, "$_" or die "error opening $_: ".$!;
    foreach(<FILE>){
	chomp;
        $dn = $1 if(/dn=(.*)/);
        push @{$hash{$dn}{$1}}, $2 if(/(^=*)=(.*)/ && $dn ne "");
    }
    close FILE;
}

if it is case2, I don't know why you would need to add the same line more than once. if you need the number of occurances try :

 
foreach(<FILE>){
    chomp;
    $dn = $1 if(/dn=(.*)/);
    $hash{$dn}{$_}++ if(/=)/ && $dn ne ""); 
}

for the inner loop. or use an array for the "name=value" entries:

foreach(<FILE>){
    chomp;
    $dn = $1 if(/dn=(.*)/);
    push @{$hash{$dn}}, $_ if(/=/ && $dn ne "");
}

hope that helps.