Hash function - uniformity / strong universality
Arnab provided the answer. The family $\mathcal{H} = \{h_i : i \in N\}$, where $h_i(x) = i$ for all $x \in U$, is uniform but not pairwise independent. Similarly you can come up with families which are pairwise but not $3$-wise independent, and so on.

To give a simple example, let $X,Y$ be two independent uniformly random coin tosses. Each of the possibilities $(H,H),(H,T),(T,H),(T,H)$ has the same probability. Now let $X'$ be a uniformly random coin toss, and let $Y' = X'$. Now it is not true that each of the four possibilities of $(X',Y')$ has the same probability, but each of $X',Y'$ by itself is a uniformly random coin toss.