The easy answer to whether or not Carbon Nanotubes (CNTs) can cause mesothelioma similiar to asbestos exposure is a quick "maybe."

The truth is there have been some studies that have shown that the longer, thin nanotubes can cause lesions and scars in the lungs of throats, similiar to what happens from asbestos exposure. With other types of CNTs, no such affect is applicable. 

Asbestos exposure of course has led to thousands of deaths from exposure, especially for those people who are military veterans (especially those in Navy) and people in the construction industry. To avoid another disaster like this,  it could be said there has been an overreaction amongst the cancer and science community.

What appears to be true is that end users have nothing to worry about, which is a big relief for 99.9% of the population. Rather, it's those people who manufacture and dispose of the product that could have something to concern themselves with. 

CNTs are commonly used in products like tennis racquets , baseball bats and computer wiring--revolutionizing all of the said items.

Now, the reason I feel we need to be cautious are the emerging unique benefits of nanotubes in medicine. Scar tissue prevents nerve regeneration for persons suffering from spinal chord injury. In presence of nanotubes, nerves seem to re-grow. Bone loss occurs in
prosthetic devices at the interface to metals (e.g. hip bone replacement). Nanotubes appear to stop the bone loss. The is an absolute first in medicine to provide this cure. It seems to work...but the side effects have to be studied.