I added a simple code <!--more--> at where I want to break the page. It didn't work.
It used to work. My old posts work as it were set up on the same website.

versions? url? using excerpts? where did you put the <!--more-->?

AT ver. 3.73
WP ver. 3.31
www..com
Edit post use HTML inserts break page code at where the break is. save it. It works. Last couple posts, I found out it was not working when I tried to set where to break the page. If you archive the older post, you should see at the end of the post showing 1,2,3...
The new post was appearing as 'continue reading.....'
Another strange thing was happened, there was a 'insert social media' button on dash board (which I don't know how it came from), I clicked on it, it gave me an option where I want to put the icons, I selected to put it on post page. the icons works. Today, I opened the dash board, 'insert social media button' was gone. I suspected it came with the source link. At the end of 'continue reading ... post', link social media icons are still there.

Disable all your plugins and see if the more works.

If you have things showing up and you press buttons but don't know the ramifications that seems to me like a recipe for disaster.

I hope you are doing regular backups.

Also, what do you have the excerpt length set to? Does the more fall before or after that number of words?

I am glad you asked. I don't set the break page by numbers of word, rather by the content and by rows, no specific rule. I am not planning to upgrade to WP 3.4, until all the plug-in are known working ok with the new version.

please do ALL the following

1) Disable all your plugins and see if the more works. What is the result?
2) what's the value at ato->Configure EXCERPTS->Excerpt length
3) what's the value at ato->Configure EXCERPTS->Posts or excerpts on HOME page
4) what's the value at ato->Configure EXCERPTS->Custom read more
5) what's the value at ato->"Read More" tag->Read More
6) what is the title of a post with the issue?
7) Edit that post using the HTML editor - copy the entire post and paste it to a text file. Include that file with your reply (use the paper clip icon.