A pixel is the smallest unit value to render something with, but you can trick thickness with optical illusions by modifying colors (the eye can only see up to a certain resolution too).
Here is a test to prove this point:
div { border-color: blue; border-style: solid; margin: 2px; }
div.b1 { border-width: 1px; }
div.b2 { border-width: 0.1em; }
div.b3 { border-width: 0.01em; }
div.b4 { border-width: 1px; border-color: rgb(160,160,255); }
Some text
Some text
Some text
Some text
Output

Which gives the illusion that the last DIV
has a smaller border width, because the blue border blends more with the white background.
Edit: Alternate solution
Alpha values may also be used to simulate the same effect, without the need to calculate and manipulate RGB values.
.container {
border-style: solid;
border-width: 1px;
margin-bottom: 10px;
}
.border-100 { border-color: rgba(0,0,255,1); }
.border-75 { border-color: rgba(0,0,255,0.75); }
.border-50 { border-color: rgba(0,0,255,0.5); }
.border-25 { border-color: rgba(0,0,255,0.25); }
Container 1 (alpha = 1)
Container 2 (alpha = 0.75)
Container 3 (alpha = 0.5)
Container 4 (alpha = 0.25)