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In this question, fizz and Shiny Hunter Luna kindly explained that a Pokemon can gain stat boosts from EVs without leveling up, and if a Pokemon has a lot of EVs, it goes into the stats a little more every time the Pokemon levels up. First, my Pokemon are only getting stat boosts when they level up. Second, the previous explanation would imply that if a Pokemon had a lot of EVs in a particular stat, every time it leveled up, that stat would get a little higher every time it leveled up. However, when a Pokemon's EVs are maxed, the first thing that I notice is that the Pokemon's stats are increasing as if there were no EVs. Can I get a better explanation?

I dont really understand the question, but the stats go up once each 4 EVs, but it doesnt tell you "<pokemon>'s Attack went up by 1!" Or anything
When a Pokemon levels up, it shows how much was gained from leveling up.
Yes (filler)
I do look at summaries quite often.
btw why the filler?
Because a comment has to be 6 characters long or it wont let you post, so i just added a filler since i had nothing else to say
it tells you how much of a stat is added after level-up, it does not include EVs. EV's aren't earned by level up, they are earned through battles
Iirc the "get EVs that boost stats automatically" was put in the newer gens, so if youre playing somerhign that isnt that recent it might not have the feature
oh. (filler)

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In the current gens your Pokemon's EV's can effect their stats even without leveling, for instance by feeding your level 100 Shiftry a carbos it will still gain 2 or maybe 3 in speed. There will be no announcement for this boost however. Now lets take the same shiftry and make it 10 years younger, back in gen 3 and such that shiftry would've NEVER gotten that carbos because of it's level, which was 100, so it could never level up to collect the boost

You're likely playing an older game if your EVs are only popping in only when you level up (+7 in attack and other unreasonably high raises) and the reason your Pokemon seem to be gaining no evs at higher levels is because they probably hit their EV limit, and can no longer even obtain EVs.

It would be nice to know what game you're talking about and how you're raising your Pokemon so we can give a better answer

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Uh can you explain where you got this info? Because EVs work the same way in Gens 3-6. The system was nothing like it is today in Gens 1 and 2. As my answer in the other question explained, if a Pokemon gets enough EVs, it may gain stat points without levelling up, so there's no such thing as "collecting" points when you level up. EVs are just worth more the higher a Pokemon's level is, since the stat formula multiplies the EVs by the Pokemon's level.
I believe DemonFlygawne is correct. Before Black/White was released Pokemon EV's were applied at level up. Starting with Black/White EVs were applied at the end of battle.
"In Generations III and IV, for every Pokémon that has gained Effort Points in battle, stats will be recalculated upon leveling up, except for Deoxys, whose stats are recalculated after every battle instead."  - http://bulbapedia.bulbagarden.net/wiki/Effort_values
Starting with Black/White Every Pokemon's EV's are recalculated at the end of the battle.
Its tagged Platinum