Sigh. Here it is again. Make lots of links to this post and place them in prominent places on SDN, your school's bathrooms, the lockers, the anatomy lab - everywhere.
ERAS is nice because it can be done in bits and pieces. The only thing you need to have completed to "apply" is the CAF. Once that is completed you can submit your application to the programs you are interested in and PD's will see it. Last year I know many students interested in Peds, IM, FP got interviews with nothing more than this and their USMLE scores sent to programs. The earliest you can certify the CAF and submit it is Sept. 2.
After you have submitted the CAF the other pieces can be sent one by one as you complete them. The way it works is that programs have computer software that constantly updates your file for them from the ERAS database(by constantly I mean 1-2x per month). Usually you have to have all your material (LOR's, Scores, Statement) in by the program deadline which is usually about Nov. 1-2 when Dean's letters come out.
It is possible to send different Statements to every program, and different letters to every program, or the same ones to all of them. It is possible to send one statement, retract it, and send a different one. Same for letters. So overall the system is pretty flexible.
So to answer your question directly, in competitive programs (Rads, Anes, EM, Cat Surg) I would say people generally have their CAF ready to go on Sept. 2 so that "their name is on the list." The rest of the material follows during Sept. and hopefully is complete by early Oct. You can usually add letters right up to the programs' deadlines. Sometimes the first time they will see a complete file is on your interview day, but they generally don't expect a complete file until Nov. 2.