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); }

<div class="b1">Some text</div>
<div class="b2">Some text</div>
<div class="b3">Some text</div>
<div class="b4">Some text</div>

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); }

<div class="container border-100">Container 1 (alpha = 1)</div>
<div class="container border-75">Container 2 (alpha = 0.75)</div>
<div class="container border-50">Container 3 (alpha = 0.5)</div>
<div class="container border-25">Container 4 (alpha = 0.25)</div>