If an addiction can be identified as an activity which one is unable to control their impulses and are willing to risk everything in order to get their fix.... then I would say it's possible to have a sex addiction.
However as a society we tend to frown upon "sin addictions" such as porn, sex, and gambling as being "legitimate". Nonetheless there are scientific studies that have been conducted which show similar brain activity as those who are addicted to tobacco, alcohol, or caffeine.
The primary reasons why we have such a difficult accepting "sin addictions" is because they do not involve (ingesting) a substance and are viewed as nothing more than selfish impulsive behavior. Last but not least the bulk of the pain is felt by those who loved and trusted them. If they are diagnosed as being "addicted" we see it as a convenient story to (excuse) their bad behavior. Naturally not (everyone) who professes to have an "addiction" has one. And even if such an "addiction" could be proven most likely it would not change the minds of those who view these actions as being a "deal breaker".
Today more and more (behavior) is being diagnosed as addictions or a form of mental illness. These include "food addiction", "shopping addiction", "hoarding" and even shoplifting. In fact there is an organization called "Cleptomaniacs And Shoplifters Anonymous". However none of these "addictions' cause the same amount of heartbreak as discovering one's mate is having sex with others.