OK, following up with this (I don't have my Kindle at hand at the moment, so I'm trying several options):
1. I tried sending a page from a website (seriouseats.com) with a recipe to my free.kindle address. I got a response saying the personal document service found nothing to convert.
2. I tried sending the text from a website in an email to my free.kindle address, with the same response.
3. In Google Chrome, I saved the page (the print version page for the same recipe) onto my hard drive. This resulted in an HTML file and a folder with supporting images and css/js. I emailed only the HTML to myself (as I'm only interested in the text, not the ads, logos, images, etc.). So far no error; still need to confirm the formatting on the Kindle when I get home.
It seems @KindleWorm is on to something with saving her documents in Word. I will report back with more info as things progress!