**Answer:** Yes.

Take a piece of fruit from the box labelled apples and bananas.

If an apple is taken, then this box is the apples box (as it can't be the apples/bananas box as all of the boxes are mislabelled). This then means that the bananas box is labelled apples, and the apples/bananas box is labelled bananas. Similarly, if a banana is taken, this box is the banana box, the apples box is labelled bananas and the apples/bananas box is labelled apples.