Wordpress | Blank First Line in RSS feed | Broken RSS feed

We've just had this happen to TWO sites: The first line of the RSS feed comes out blank, meaning that the RSS feed is broken. Feedburner can't cope, so the RSS subscriptions, Twitter feed, and e-mail subscriptions all disappeared down a black hole. This happened after a couple of major site revisions, involving new plugins. First up, we tried the "fix-rss-feed" plugin. Nope. Then the "whitespace" fix. Nope. Removing blank lines from wp-config. Nope. Hmm.. The solution is the obvious, boring one: disable ALL plugins, and re-enable one-by-one until the RSS feed breaks. Turns out it was nothing to do with the new plugins, it was "Advanced Tag List", which I think has just been updated.. It just goes to show - you can never be too careful. With muktiple plugins by multiple authors, this IS going to happen. Anyway - everything is again rosy in the garden.

RSS Error: Blank Line(s) at start of XML - Solution.

I noticed that suddenly I wasn't getting my daily e-mail message from one of my sites.
Now, these messages are generated automatically via feedburner, so it's either feedburner, or gmail, or the actual site RSS feed.

Checked GMAIL to make sure the messages weren't ending up in the SPAM. Nope.
Checked Feedburner. Stats looked OK, but "REACH" had dropped to zero about four days ago.
Damn. Feedburner handles the twitter feed AND email subscriptions.

Further investigation showed that the RSS feed was no longer valid. Blank lines at the start of the XML.

Tried the two main solutions:
fix-rss plugin. Nope.
Then the whitespace remover thing. Nope.

Bit more research, and then I came across a post that suggested looking in the THEME's functions.php file. Alarm bells! I edited that about four days ago..

Loaded it up, and it was a mess. I must have been asleep when I edited that.
Re-loaded the file from backup. (You DO have a backup, right?)
And away we go. RSS is back. Feedburner is back.

RESULT!