The "adventuring day" XP seems intended to budget for 6-8 encounters, but comes in about an encounter under.
More than one encounter under, if you stick to medium/hard encounters. I no longer have my table handy but I believe the typical level can fit in about 4.5 encounters that are midway through Medium difficulty (i.e. the average of Medium and Hard thresholds). There are a couple of levels where it goes as high as 6 encounters, and a few there are more in the 3-4 range. There are none that are as high as 7 or 8.
The "6-8" meme made more sense before they revised the difficulty guidelines, back around Basic 0.2. Back then, the breakpoints were ceilings, not floors, so what today is an easy/medium encounter would have been a medium/hard encounter back then. If you do the math using
those guidelines, you'll find that you actually can fit 6-8 encounters in.
Unfortunately, when they updated the difficulty guidelines and then printed them in the DMG, they did
not update the accompanying text blurb saying that "most adventuring parties can handle about six
to eight medium or hard encounters in a day," even though they had changed the definition of "medium or hard encounter."
[Note also that back then there was such a thing as "Deadlier than Deadly" difficulty, which Kobold.com used to call "Ludicrous" difficulty.]
Concrete example: if you look at Basic 0.1 pages 56-58, there's a Hard encounter given as an example encounter between four PCs (three level 3, one level 2) and four hobgoblins. That consumes 800 out of the 4200 XP budget for the day (3*1200 + 600, per table on page 58), leaving 3400 XP left. If you distribute those 3400 XP evently between six other encounters for a total of seven encounters, that gives you one Hard encounter (hobgoblins, 800 XP) and six more barely-Hard encounters (whatever else, 566 XP). That's because a Medium encounter can be at most 550 XP (3*150 + 100) and a Hard encounter can be at most 825 XP (3 * 225 + 150), according to the table on page 56. So we see that "six to eight medium or hard encounters in a day" held, back then.
Contrast that with today's DMG rules. (I'm AFB so I'll refer to Basic 0.5 instead here:
http://media.wizards.com/2016/downloads/DND/DMBasicRulesV05.pdf but they are the same rules.) Now the example Hard fight is of a bugbear and three hobgoblins against the same party of three level 3s and a level 2. Due to the addition of a bugbear, it's 1000 XP, which crosses the new Hard threshold of 825 XP. The adventuring day budget hasn't changed, so we've still got 4200 XP total to spend, and 3200 XP to split between six encounters. That gives us 533 XP per encounter, which according to the new difficulty table on page 56 means each encounter is Easy (just shy of the 550 XP threshold for Medium).
The upshot is that whereas Basic 0.1 would have given you seven Hard encounters in a day, Basic 0.5 or the rules printed in the DMG would give you one Hard and six Easy encounters. Both versions preface the "Adventuring Day" XP table on page 57 with a text blurb stating that "Assuming typical adventuring conditions and average luck, most adventuring parties can handle about six to eight medium or hard encounters in a day", but that's a holdover from Basic 0.1. In 5E as the DMG actually published, that statement is no longer compatible with the table it's introducing.
In both cases, there
never was any expectation that you have six to eight encounters a day. The expectation was that you don't exceed your adventuring day budget and accidentally TPK the party. 5E's design parameters are built to handle two or three hard/deadly encounters per day just as readily as six to eight easy/medium encounters.
What I believe gives good diversity is when you can also fit in some easy encounters that are over quickly, but matter mechanically. Adventuring days should vary (using days in a non-calendar sense here). Notionally, the XP budgets per DMG guidelines work well for that. Actually, they probably work fine for inexperienced or casual groups. Not as fine for groups that are more on top of the mechanics.
It would have been helpful if they'd given 2 or 3 columns for adventuring day XP. Novice, Standard, Elite perhaps. Tomorrow I'll have a shot at working out what those budgets should come to!
Also, risk tolerance is important. Even novice players can easily handle 150-200% of the standard adventuring day XP; but sometimes their PCs die, and some groups require a 100% chance of PC victory in order to have fun, which means those groups will have to stick more closely to the standard values.
IMO the game is at its best when the PCs are outnumbered and outgunned but not outthought; I like pitting e.g. four 9th level PCs are up against six CR 6 Chasmes and a couple of CR 17 Goristros. DMG guidelines tell me that that encounter is ludicrously difficult (124,500 XP when the Deadly threshold is 9600 XP) but my experience tells me it is about right for a couple of hours of fun. I'd love to be one of the PCs in that party, especially when they collect all the DMG-generated treasure associated with such monsters.