Black neon tetras, arent actually tetras. So I dont think they will shoal together. As someone as already said there is a size difference and that alone would most likely mean they wouldnt.
if you want something to school with neons, try get something familiar size and colour. i don't know about cardinal tetra's but they are most certainly more or less the same colour. i had white clouds that schooled with neons, probably because of the red tail.
i have got 10 different types of tetra (in groups of 8) and the only ones that shoal together are the neons and cardinals. and that was only after i introduced some angles. this is in a planted 300ltr.
I have 6 Black Neons and 6 Neon Tetras in my tank ( 29 Gal) and have not noticed the two types shoaling together.
The shoal by type Neons with Neons and Black Neon with Black neon.
This said when I had the 6 Black Neons and only one Neon Tetra the Neon shoaled with the others untill I added the other 5 Neons and then it shoaled with its own type.