This most likely has to do with the dot pitch on your particular computer's screen. For example one can watch the thickness make the plot "disappear" on either computer by changing it in a Manipulate
:
Manipulate[
Plot[x, {x, 1, 2}, PlotStyle -> Thickness[1/x]],
{x, 10, 10000}]
On my MacbookPro the line becomes essentially invisible at around 1/9000...