Your problem is here:

while (my $line =<FILE>) {
    my ($one,$two,$three,$four,$five) = split "\t",$line;
    if ($three eq $checkthree) {
       print "<p>With matching parameters I should be able to see this
+ </p>";
       last;
    } else {
        print "<p>If I put the else statement in it ignores the fact t
+hat the parameters match - and comes here!  (But I don't know why).</
+p>";
        last;
    }
}
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This is what's happening: You're getting to this while statement, and $line gets the first line from the file. You're finding that the first line of the file doesn't match $checkthree, so the else is executed, the line is printed, and the while is exited. To fix this, try some like the following.

my $matched;
while (my $line = <FILE>) {
    my ($one,$two,$three,$four,$five) = split "\t",$line;
    if ($three eq $checkthree) {
        print "<p>With matching parameters I should be able to see thi
+s </p>";
        $matched = 1;
        last;
    }
}

unless ($matched) { # this is like the else
    print "<p>If I put the else statement in it ignores the fact that 
+the parameters match - and comes here!  (But I don't know why).</p>";
}
[download]

elusion : http://matt.diephouse.com