First shorter does not mean clearer (nor better), but it is possible to shorten your code to something like so:

#!d:\perl\bin\perl.exe

$pth = "\\\\127.0.0.1\\bex";
opendir(Directory, $pth) or die "Cannot Open Directory";
@contents = grep {
          m!^[Bb].+\.txt$!  #assuming .txt ending
          and
          -M "$path\\$_" > 1
         } readdir(Directory);
closedir(Directory);
[download]

A couple of other issues though. You do not need the splice as your regular expression would have eliminated it anyhow. and second I changed your regular expression because if you want to match a period in a regular expression you can do it one of two ways backwack it (i.e. \. )or in a character class (i.e. [.] ), the .txt in your regular expression would also match anything that ended with txt preceded by any character.

update:A shorter solution for the whole thing (untested):

#!d:\perl\bin\perl.exe

my $pth = "\\\\127.0.0.1\\bex";
opendir(Directory, $pth) or die "Cannot Open Directory";
while ( $_ = readdir(Directory) ) {
  if ( m!^[Bb].+\.txt$! and -M "$path\\$_" > 1 ) {
     open ( FILE, "$path\\$_" > 1 ) or die "$!";
     while ( my $line = <FILE> ) {
       print $line;
     }
     close(FILE);
  }
}
closedir(Directory);
[download]

-enlil