Yes, likely letters go out to very competitive applicants. Each school only probably sends 200~300 likely letters, but 200~300 do make up a hefty portion of the class. Harvard sends 300 likely letters starting mid-February (not necessarily all at once), and admits around 2,000 students. That is still 15% of all admits.
To my knowledge, all Ivies except for Princeton send likely letters to non-athletic applicants. Other top tier schools like Stanford, Duke, and Chicago practice likely letters as well. Most go out mid to late February, but some (like Brown) are sent during early March. Therefore, there are still likely letters waiting to go out. At 5 PM EST today, Columbia sent out its first batch of likely letters, and will send out another early March. In my opinion, the primary reason the distribution is divided like this is because they haven’t read all the applications in committee yet.
Here is a sample likely letter from Brown:
https://www.quora.com/What-did-Kalvin-Lam-do-to-get-into-Brown
One student I know got a likely letter from Harvard in the mail dated March 18. He used to live overseas in Japan (American abroad), so he got the letter a week later on March 27 or 28. I think he knew beforehand though because it was in his portal as well. He was waiting to make the more “official” announcement.
Very random and rare indeed.