We’re not talking about a research encounter. Look at the image that I just posted. This is new. We are talking about a reward for field tasks.
THRobinson mentioned exactly what’s happened to me. I can now hold 4 tasks at a time, claimed or not.
I have 11 researchs pending. Easy way to get a pókemon when im away from any pokestop and don´t have a field research with it.
If you run or cancel and encounter with a field research reward, it goes there, stacking for later catch. Been playing since March, 2018 and it was always there…
Once I get 3, finished or not doesn’t matter, that’s it… I can spin stops until blue in the face and I’ll never get more than 3 total.
Side note… when my bag is full I can’t spin stops and get more tasks either, which makes no sense, full bag or not I should still be able to get tasks. Unrelated… but annoying.
I said that you can hold four Researches at a time because you didn’t catch a Research encounter. And now I want to end this off-topicness.
I have 3 finished and unclaimed and cannot get a 4th. Never could. I almost always have all my tasks ready to claim 2-3 at a time. I’ve never gotten a 4th task.
I have 3 finished right now. After work I’ll go spin a few stops and double check.
Or… are you saying you have all 3 finished, you try to catch a Zapados, Entei or whatever and fail… THEN you get the 4th slot?
Open one of those unclaimed Field Researches that shows a grass with a question mark symbol, and run from the encounter. You’ll see that a fourth slot appears.
Did not see this before, but it doesn’t work with the Breakthrough Box (that thing that gives you a Legendary, e.g. Zapdos, Entei). @THRobinson
Ah I getcha…
… can’t say I’ve ever run from one to see that happen.
… ok, now that that’s cleared… March 1st at 4pm and another Zapados… what a rip.
Sadly I don’t know why you got a Zapdos… 
Had a bunch of them but managed to get a Shiny with a 93IV, plus the CP was higher than the rest so, it made the decision of which to keep pretty easy, though, not sure if worth the dust to power up.