This doesn't answer John's question, which is about textareas, but the reason is because you're supposed to URLencode things that go in URLs. Internet Explorer and Opera aren't complying with the standards. URLs (including querystrings) aren't supposed to contain spaces. A browser will itself URLencode any data entered into form fields, but if you slip it directly into the URL you'll have to encode it yourself. After all, what would happen if $name had an & or = sign in it?
I presume you meant to write
echo "<a href=\"showme.php?name=$name\">$name</a>";
because the precise example you give is simply broken.
On the subject of John's post, are you POSTing your form data? GET-posted data has a limit on size... It's an obvious question, but I've managed to trip on it once.
Otherwise, I'd like to see some code so that I can make a more informed diagnosis - the <textarea> tag in question and the bit in the script that takes it in. Have you tried echoing the textarea's supposed value at the very start of the script (if that's feasible)?