In the Hell's Kitchen OVA set between episodes 2 and 3, we learned from Yuri's event application form that the deadly picnic took place on May 3.
Furthermore, Yuri allowed for one week of preparation for the picnic prior to the Golden Week. Taking that into account and given that the series of events connecting the first two episodes and the second OVA episode were tightly packed, it is probably safe to say that Otonashi woke up in the Afterlife World sometime around the start of the first term of the Japanese trimester system, maybe even on the first day of the fourth month as an April Fool joke by God.
There is no set date for the Ball Day (球技大会), but with its conditions being similar to the day of the regional game final that would decide if Hinata and his team get to participate in the Summer Koushien, it should be set in late July during the Dog Days, or not, if the anime followed the logic explained in the next paragraph.
Throughout the anime, everyone wore the same winter uniform, unlike in the Angel Beats! Heaven's Door manga where they actually switched to the summer uniform once. As the dates of the seasonal uniform switch in Japan (June 1 and October 1) are strict and almost universal, there is little doubt that the anime didn't show anything that occurred between these two dates.
Following episode 4 and preceding episode 5 is the Stairway to Heaven OVA episode. Otonashi and his comrades organized the Sports Day that normally occurs in September/October. Then by episode 5, the students were taking their second midterm exams in early/mid October per usual.
November largely overlaps with the tenth month of the lunar calendar, the Kannazuki (神無月), or the "Month with/without Gods".
The 無 character, which normally means "absent" or "there is not", was here probably originally used as ateji, that is used only for the sound "na". In this name the na is actually a possessive particle, so Kaminazuki means "Month of the Gods", not "Month without Gods" (Kaminakizuki), similarly to Minatsuki, the "Month of Water".
Whether God exists or not is still debatable, like how it is with the na in Kannazuki. Interestingly, Naoi proclaimed, in episode 6 during the "Month with/without Gods", that there was no God and that he is God. Maybe God is a Schrödinger's cat in the tenth month of the lunar calendar.
From mid-November to early December is the "rainy season of the Camellia". (The only times it rained in the anime were in episodes 6 and 9.) We saw the blooming Camellias in episode 7:
We also saw Kanade weeding the Chrysanthemums japonense, which typically flower in November annually.
I couldn't pick up any seasonal cue in the later episodes, so I'm less confident with the time of the events happening in episode 8 and onwards. It was not clear how long Kanade stayed in a coma, but could it be that she woke up on or around the day Hatsune died (Christmas Eve)? I couldn't help make this assumption because of the similarities between Kanade and Hatsune.
And then there is Matsushita's mountain training. It is possible, but highly unlikely, especially for a person like Matsushita who is not too fat, to lose 10 kg in a week. Judgment by the eyes (my eyes) tells me that he had lost 20–40 kg by the time he returned from the mountains in episode 12. That equates to a 3–6 weeks minimum time gap between the beginnings of episodes 10 and 12. However, it is far more likely that he was away for longer than that, probably 1–3 months to coincide with the time of the graduation, which is in March. I would wager that the anime ended with the end of the third term in March, excluding the epilogues.