CSS generated content can solve this for you:

div {
  position: relative;
}


/* Main div for border to extend to 50% from bottom left corner */

div:after {
  content: "";
  background: black;
  position: absolute;
  bottom: 0;
  left: 0;
  height: 50%;
  width: 1px;
}

<div>Lorem Ipsum</div>

(note - the content: ""; declaration is necessary in order for the pseudo-element to render)

I have another possibility. This is of course a "newer" technique, but for my projects works sufficient.

It only works if you need one or two borders. I've never done it with 4 borders... and to be honest, I don't know the answer for that yet.

.your-item {
  position: relative;
}

.your-item:after {
  content: '';
  height: 100%; //You can change this if you want smaller/bigger borders
  width: 1px;
  
  position: absolute;
  right: 0;
  top: 0; // If you want to set a smaller height and center it, change this value
  
  background-color: #000000; // The color of your border
}

In CSS3, you can try to (ab)use the max css function, if your browser supports it.

border-width: max(1px, 0.1em);
border-style: solid; 
border-color: black;

Unfortunately this awesome CSS3 feature isn't supported by any browsers yet, but I hope this will change soon!

But in CSS2 – no, you can't.
However, you can use JavaScript/jQuery to loop through all elements and increase the border size to 1px.

But this will eat so much performance your browser is gonna crash if you have too many elements on your page (e.g. a table with more than 50-100 rows).
So in other words, no it's not possible.

$("[id$='ReportViewerControl']").find('*')

    .each(function () {

        if($(this).is('#ParametersRowReportViewerControl *')) 
            return;

        //console.log("Processing an element");
        //var cls = $(this).attr("class");

        // Don't add a border to sort-arrow
        if ($(this).is('img')) {
            return;
        }

        var anywidth = $(this).css('width');
        var anywidth = parseFloat(anywidth);
        //console.log("anywidth: " + anywidth);


        //var lol = $(this).css('borderLeftWidth');
        var blw = $(this).css('border-left-width');
        var brw = $(this).css('border-right-width');
        var btw = $(this).css('border-top-width');
        var bbw = $(this).css('border-bottom-width');

        var borls = $(this).css('border-left-style') == "solid";
        var borrs = $(this).css('border-right-style') == "solid";
        var borts = $(this).css('border-top-style') == "solid";
        var borbs = $(this).css('border-bottom-style') == "solid";

        var blw = parseFloat(blw);
        var brw = parseFloat(brw);
        var btw = parseFloat(btw);
        var bbw = parseFloat(bbw);

        //parseInt($(this).css("borderRightWidth"))
        //console.log(parseInt($(this).css("borderLeftWidth")));

        // UpdateLock = true;


        // Set width to 1px where 0px
        if (anywidth == 0)
            $(this).css('width', '1px');

        if (borls && blw == 0.0 || (blw > 0.0 && blw < 1.0)) {
            //console.log("setting border width");
            $(this).css('border-left-width', '1px');
        }

        if (borrs && brw == 0.0 || (brw > 0.0 && brw < 1.0)) {
            $(this).css('border-right-width', '1px');
        }

        if (borts && btw == 0.0 || (btw > 0.0 && btw < 1.0)) {
            $(this).css('border-top-width', '1px');
        }

        if (borbs && bbw == 0.0 || (bbw > 0.0 && bbw < 1.0)) {
            $(this).css('border-bottom-width', '1px');
        }

        // UpdateLock = false;
    });           // End $('*').each

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>