Tengo una configuración flexible bastante básica y, por cualquier razón, el div en cuestión no se centrará verticalmente dentro de su etiqueta principal. Puede ver el caso de prueba aislado a continuación.

.likeness-rank-table {
  border-radius: 3px;
  margin-bottom: 20px;
  display: block;
}
.likeness-rank-table .col {
  box-sizing: border-box;
  text-align: center;
  position: relative;
  flex: 1;
  max-width: 20%;
  min-height: 50px;
  display: flex;
  flex-wrap: wrap;
}
.likeness-rank-table .col .col-inner {
  flex: 1;
  align-items: center;
  justify-content: center;
  height: 150px;
}
.likeness-rank-table .likeness-rank-table-body {
  display: flex;
  border-bottom-left-radius: 3px;
  border-bottom-right-radius: 3px;
  border: 1px solid #a2a5a7;
}
.draggable-tags .draggable-tag {
  display: inline-block;
  float: left;
  width: 20%;
  margin-bottom: 5px;
}
.draggable-tag {
  text-align: center;
  padding: 0 15px;
  box-sizing: border-box;
}
.draggable-tag .draggable-tag-inner {
  position: relative;
  padding: 10px 0;
  background-color: #63b3ce;
  color: #fff;
}

<div class="likeness-rank-table">
  <div class="likeness-rank-table-body">
    <div class="col">
      <div class="col-inner droppable">
        <div class="draggable-tag ui-draggable">
          <div class="draggable-tag-inner">
            This Blue Box Should Be Horizontally and Vertically Centered Inside The Big White Box But It's Not
          </div>
        </div>
      </div>
    </div>
  </div>
</div>

Aquí hay un codepen para verlo:

http://codepen.io/trevorhinesley/pen/grQKPO