Here is the interesting thing about the wrong bacteria. As daize noted, it can often provide a bridge. In one of the studies done by the group that identified the nitrite oxidizing bacteria in fw tanks, they tested a bottled product which contained Nitrobacter. This is a nitrite bac that handles high levels of nitrite. This research was the paper that identified Nitrospira as the dominant nitrite bacteria in tanks. This is a variety that works best at the lower nitrite levels which are typical in tanks.
What the research discovered was that when the wrong bacteria in a bottle was used, the onset of nitrite conversion started earlier but in the end the cycle itself still took the same amount of time. And when it was done there were basically no Nitrobacter presen,t there were only Nitrospira. This can also happen with ammonia bacteria.
The big difference once can see between bacterial starters is in how much time they cut off of the standard cycle. In the above case even though the bacteria did hold down nitrite some, in the end, the cycle was not shortened. This is why I am such a big supporter of Dr. Tim's product. When used properly, it can fully cycle a tank in about one week. The same can hold true wgen using seed bacteria taken from established tanks.