It looks like Joseph O'Rourke has beaten me, but I've generated a bit more data than he has:
Let $l$ be the lower bound so that the non-diagonal entries are uniformly distributed between $l$ and 1 and the diagonal entries are distributed between $l+1$ and 2. Note that Piero's original characterization of his code was inaccurate, but I just used his code to generate the figures below.
Here is a grid of eigenvalues of such $N$ by $N$ matrices with $l=-.9$ (plots are labeled by $N$):
Here's the picture with $l=-.87$:
Here's the picture with $l=-.93$:
And here's a picture of the absolute value of the largest eigenvalue of the matrix as a function of $N$ (now the label is $l$):
Edit based on Helge's comment: Here's a picture of $\max_{1\leq j \leq N} \sum_{n=1}^{N} u_j^N(n)$ as a function of $N$ (the $u_j^N$ are normalized eigenvectors) with $l=-0.9$:
And here's a picture of the further normalized version $\max_{1\leq j \leq N} \frac{1}{\sqrt{N}}\sum_{n=1}^{N} u_j^N(n)$:
Edit 2: In response to Helge's comment asking for a plot of $\max_{1\leq j \leq N} \sum_{n=1}^{N} u_j^N(n)$ when $l=-1$:
