Solution 1:
Use the webkit only -webkit-line-clamp property for 2 lines.
.overme {
    width: 300px;
    height: 60px;
    line-height: 30px;
    overflow:hidden; 
    font-size: 30px;
    color: red;
    background: #333;
    /*The problematic part is below*/
    white-space:nowrap; 
    text-overflow: ellipsis;
}
    
.overme {
    white-space: normal;
    display: -webkit-box;
    -webkit-line-clamp: 2;
    -webkit-box-orient: vertical;
}
<div class="overme">
    how much wood can a woodchuck chuck if a woodchuck could chuck wood?
</div>
 
 
Solution 2
Use an :after pseudo element, aligned to the bottom right corner.
This only works if your text is static and you known beforehand that will overflow the container.
.overme {
    width: 300px;
    height: 60px;
    line-height: 30px;
    overflow:hidden; 
    font-size: 30px;
    color: red;
    background: #333;
    position: relative;
}
    
.overme:after {
    display: inline-block;
    position: absolute;
    right: 0;
    bottom: 0;
    content: '...';
}
<div class="overme">
    how much wood can a woodchuck chuck if a woodchuck could chuck wood?
</div>
 
 
Solution 3 - Cross-browser
This JS solution compares the height of the parent container (div) against the content text. If the text height is greater than the parent's height, then a .overflows class is added to the parent.
To test this, delete some text so that it fits all in the parent. You will no longer see the dots.
$(".overme").each(function () {
  var $elem = $(this);
  $elem.addClass($elem[0].scrollHeight > $elem.height() ? 'overflows' : null);
});
.overme {
    width: 300px;
    height: 60px;
    line-height: 30px;
    overflow:hidden; 
    font-size: 30px;
    color: red;
    background: #333;
    position: relative;
}
    
.overme.overflows:after {
    display: inline-block;
    background: #333;
    position: absolute;
    right: 2px;
    bottom: 0;
    content: '...';
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div class="overme">
    how much wood can a woodchuck chuck if a woodchuck could chuck wood?
</div>