Why don't you check your match for success? If if fails, you get the $1 from the previous iteration. That's probably not what you intended. Also, I'd use \d+ instead of \d* since we're not interested in matching tim without any following digits. It probably should be anchored to the front of the string too, by the OP's spec.

my @wanted;
opendir my $dh, $dir or die "Failed opening $dir: $!";
foreach (readdir $dh) {
    next unless /^tim(\d+)/;
    my $num = $1;
    push @wanted, $_ if $num > 14 and $num < 51;
}
closedir $dh;
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Personally, I'd wrap the whole thing in a do block and use grep, so that I don't need to track my handles explicitly:

my @tim_file = do {
    opendir my $dh, $dir or die "Failed opening $dir: $!";
    grep {
        my ($num) = /^tim(\d+)/;
        defined $num and $num > 14 and $num < 51;
    } readdir $dh;
};
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Note that when you assign a match's captures to a list, failed expressions will return undef, so here I can implicitly test for success by testing the definedness of $num.

Makeshifts last the longest.