In response to this:

I would also like it to count how many matches it makes because the input could possibly have more or less than 3 matches.

Here's how that would look using split:

while (<DATA>) {
    chomp;
    my @pieces = grep /[a-z]/i, split /[()]+/;
    printf( "Got %d pieces from '%s' : %s\n",
            scalar @pieces, $_, join( ' and ', @pieces ));
}

__DATA__
(network)(test)(ifcfg)
(network)  (test)_(ifcfg)
(foo).(bar)
(one)(two)--(three)/(four),(five)
[download]

Now, if your real input ever contains letters between a close paren and a following open paren -- e.g. "(foo) X (bar)" -- this approach will catch those as "pieces" as well. If that's a problem, go with Text::Balanced as suggested above.