Hi, I believe they can be kept together. The two have similar temperature, PH, and food requirements. I think black neons prefer PH slightly acidic, like around 6.0, which I think blind cave tetras are okay with. Blind cave tetras get 3 or 4 inches long so I think the smallest tank I would go for would be 30 gallons, especially for two schools of tetras (I'd do 6 of each).
Blind caves also like dimmer lighting and they like to eat towards the bottom of the tank so it would be ideal if you could make sure food sinks down to the bottom for them. Might be they won't be 'competing' with the black neons for food in that case, because I'm sure I remember my black neons preferring to eat from the top or middle column, back when I had them.