More pertinently, has anyone ever been both a failure and labeled as tenacious? Tenacity seems like one of those words that we use to describe successful people, making the linkage almost embedded within the word and, therefore, uninteresting.
In boxing, there are fighters who never achieve "greatness", or see much success, but are regarded as having heart. What could be more tenacious than knowing you're outmatched and still getting up to press the fight? The 09' Celtics might also be an example.
So, yes, I'd say there are people/organizations who are failures and labeled tenacious.
If that first business were successful, people fond of the buzzword 'tenacious' might label you tenacious, for toughing out the hard times in order to 'win'. Since it failed, they could keep the buzzword alive by bumping it up a level and saying that you were tenacious in pursuing the entrepreneurial dream, wisely jettisoning that failed project and learning great lessons along the way. Neither of those uses of the word would surprise me.
What would be surprising is if the word 'tenacious' were to be decoupled from success. That is, if you never hit it big but were so renowned for your tenacity that you nevertheless got press or recognition for it. Perhaps Daniel Ruettiger (Rudy, from Notre Dame fame) is the best example of someone who is recognized for his tenacity despite not achieving canonical success.*
* = at least, not in football; he is making plenty as a motivational speaker.