I've been reading over two papers recently. The first, "Why Simple Hash Functions Work: Exploiting the Entropy in a Data Stream" proves that, assuming there is sufficient entropy in a data source, weak hash functions end up producing distributions very close to truly uniformly random distributions. The other paper, "On the $k$-Independence Required for Linear Probing and Minwise Independence" gives strong lower bounds on the expected cost of linear probing lookups assuming different strengths of hash functions.
I'm having trouble reconciling the results of these two papers. The first paper proves that using 2-wise independent hash functions, if the entropy of the data source generating the elements of a linear probing hash table is sufficiently high, then the expected cost of a lookup is O(1) (and, more strongly, is equal to the bound assuming totally uniform hashing plus an o(1) term). On the other hand, the second paper proves that there exists a family of 2-universal hash functions for which the expected cost of a lookup in a linear probing hash table is $\Theta(\sqrt n)$.
These statements seem at odds with one another. I suspect that there's some technical condition from one of the two papers that I'm missing that make these bounds not conflict.
How can both of these statements be true at the same time?