I was one of the highest level players in my area. Until one guy, who initially levels up in a similar speed with me from level 20 to level 26.
But then this guy goes from 26 to 28 in one single day, and from that point he levels up with the speed of 1 level per day and he's currently level 32.
I don't know if this is possible without cheating. Hope some players with high level can answer me.
update 7/28/2016
Now two days past. This guy gains two more levels. He's currently level 34.
That means he gains 1750k XP in less than two days!
I don't think that is any close to doable without a bot. For those of you who think lure + pokemon stop + evolving + lucky egg can achieve this, I think it's probably because your level is too low to understand how hard it is to leveling up in such high levels (30 above).
Best Answer
Let's math this.
First of all, are you 100% confident in the data you've shared with us? Are you sure it isn't two different people with similar nicknames etc? I'll assume you're correct for now, but bear in mind that it's easy to get this stuff wrong.
http://www.serebii.net/pokemongo/trainer.shtml gives xp values for the following levels:-
for a total of 2.1M XP over the course of six days. That 500k for level 32 looks suspicious to me, but we'll go with it for now.
Let's be charitable and assume your rival is playing 8 hours per day, and using lucky eggs for the entire duration (at a cost of about $40). This means he needs to sustain about 45k XP per hour (or about 22.5k XP before the lucky egg bonus).
Currently the most effective (legitimate) way to gain XP is to mass-evolve pidgeys/caterpies/weedles. It costs 11 candies to evolve one (12, and you get one back). You get 3 candies for a catch, plus one for transferring (either pre- or post-evolution). This means that you can evolve four pidgies, caterpies, or weedles for every eleven you catch.
Thus, one catch gives you a maximum of:-
= ~400 XP per catch. Doubled for a lucky egg is 800 XP. Realistically you aren't going to get an excellent curveball on every throw, but this certainly gives us a reasonable upper bound.
45k XP / 800 XP per catch = around 55 catches per hour.
In order to have gone from level 26 to level 32 over the course of six days, your rival would have to have done at least the equivalent of catching roughly one pidgey, caterpie, or weedle every minute, eight hours per day.
You may draw your own conclusion.
As per the comments, you can see there are some situations where you can prepare in advance to achieve very high XP/h over short periods. In particular, the famous evolve-pidgies-with-a-lucky-egg trick peaks out somewhere around 150kxp/h (for 30 minutes...). However these tricks all rely on a previously-collected stash of pokemon/candies, and so cannot be used to sustain high XP/h indefinitely.
You'll notice I've stopped short of accusing your rival of cheating. It's not for me to say whether or not he's capable of throwing 100% excellent curveballs, or whether or not he's found a nest that genuinely lets him catch a pidgey a minute (or whether he's playing for 18 hours a day and thus "only" needs to catch a pidgey every 150 seconds...) The math here is simply to give you an idea of the level and quality of activity necessary to sustain the claimed XP rate, and thus to empower you to make your own decision about whether or not you think it's more likely that a) your rival is actually sustaining that level of activity, or b) your rival is cheating.
Response to update
To have gone from level 32 to level 34 in 2 days would require our rival to be catching a pidgey/caterpie/weedle with 100% excellent curveballs under the influence of a lucky egg once per minute for eighteen hours per day (800 XP/m * 60 * 18 * 2 = ~1.75M XP)