I started by drawing possible solutions by hand. I found one in about 15 minutes.

I then wrote several scripts that randomly tried to find a solution. I ran 3 of them in parallel, one of them finished in 74 minutes. Here's the program:

I used gnuplot to visualise the solutions. You can see a short video showing the solutions covering more than 14 dots here. The final solution is shown in the end. Note that the algorithm is really stupid, the fifth picture can be extended into the solution easily.

To generate the pictures, I used the following shell script:

#!/bin/bash
i=0
while read l ; do
    l=${l#*: }
    perl -pe 's/[^-\d.]+/(" ", "\n")[++$i % 2]/ge' <<< "$l" > 2
    gnuplot -e "out='$(printf %04d $i)'" 4x4-16.gpl
    echo "$l"
    ((++i))
done < <(sort -u 1)
[download]

and the following gnuplot script:

set xrange [-2:6]
set yrange [-2:6]
unset border
unset xtics
unset ytics
set term png
set output "4x4-16-" . out . ".png"
plot '2' using 1:2 with lines notitle, "4x4-16.grid" using 1:2:(0.1) w
+ith circles notitle
[download]

Where the "grid" file just contains all the coordinates of the points to connect.