There is no way to ensure that the whole name will be ALWAYS displayed. You can only tweak the pesentation a little bit. A few ideas:
1. Activate the chart and press Ctrl-Shift. You'll be able to resize the internal elements of the chart.
2. Using diagonal labels might allow more space than vertical labels.
3. Consider switching to Horizontal bar chart instead of the Vertical.
Another option is to remove the label from your chart (untick in label in Dimensions) and have your label in your caption. If you have rediculously long names you can choose how many lines you would like to have on your Multiline caption.
Another option (depending on your design) could also be to place your label in a text box above your chart.
Although it seems a bit less elegant solution - you gain space in your chart (bars make more sense) and you gain lots more space - if you have rediculous long names.