I must see this kind of question every week. Why try to be infinitely clever using a regular expression? Just let split do the work.

system("find /clear/pp_00/ngbb/ -name '*.txt' > CCfiles");
open (OUT , "CCfiles" ) or die " can't open CCfiles\n";
while (<OUT>)
{
  #  Original code

  #  $_ =~ m/^//\w+////;
  #  print "the someName dir is $_\n";

  my @Result = split ( /\//, $_ );
  print "the someName dir is $Result[1]\n";
}
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No doubt there will also be Monks suggesting you use the File::Find module instead -- it's a cleaner solution to running a command through system() to get the directory scan results you want.

--t. alex
but my friends call me T.

Update: My 300th node! Thanks everyone!!