You could do it, sure. Unless you had some funky water I would not see a point. Just as hobbyists we add back to the RODI water by dosing (salt, iron, calcium, etc.) so do commerical bottled waters that are either using RODI/distilled water. Depending on consumption you could end up with mineral deficiencies and associated health complications. Though would a couple cubes be harmful, not really.
DI resin is not designed for the food industry so it does not carry an NSF Certification. For this reason alone, we do not recommend it for drinking water use. Drinking solely RO/DI water can potentially cause osmotic imbalance in people that have very poor diets or have certain medical conditions.
For healthy people with normal diets, drinking RO/DI water is almost never an issue.
For best practice, we recommend teeing off the RO and collect your drinking water before the DI. We offer remineralization filter cartridges that can be easily added inline to add minerals back to the water and improve taste.