Its been my experience too that some Japanese will react that way, but I think that's more an expression of modesty/failiure to understand the test. Native Japanese should get 100% on the listening and should have little problem with the reading section. Most should be able to ace the kanji section as well, since no writing is involved and it only tests reading skills. Compared to the 漢字検定, the JLPT is a walk in the park, and high number of high school graduates should be able to complete level 2 of that test, when the kanji section of JLPT 1級 would be more equivalent to level 5 of the 漢字検定 in terms of difficulty (although not easily comparable).
The only part that could be tricky is the grammar, a lot of the stuff on level one (I've heard) is seldom if ever used. But even so, just from having a good feel of the language, educated guessing should make it not too bad.
I also imagine that many Japanese, just to make other feel better about the difficulty of the test they are doing, say that of course it would be 無理 for them. Its just an extension of the すごい、挨拶が出来るなんて feeling.