Logistic Regression Abuse?

I have a 7 products made batchwise on a product line. An event occurs on the product line that has several subclasses. The drivers for this certain event to occur on the line are similar across all of the products. This event may occur up to four times for each batch made. For it to occur more than once, some of the times it means the initial event failed to produce its intended outcome and the event must be repeated. Other times it means a time frame has elapsed since the last more robust event has occurred and regulatory action says these more robust events must be repeated within a certain time frame. That time just happened to occur during my production run (production is 24/7).

The event and its repeated events and more robust events are subsets of a parent class of events. I would like to model the frequency of the events without concern for what type of subclass of event occurred given a certain input variable.

I thought I could logisticly regress the event in tiers. So get the probability curve to the question did the event occur one time or not feeding data of a common variable from each of the seven products into a multiple logistic regression model in R. Then get the probability curve to the question did the event occur two times or not? And so on for up to four times.

I don't really need to quantify dependence on the events to one another (that's another kettle of fish I'll tackle another time), I just care to know the probability of the event occurring 1x, 2x, 3x or 4x during a production run on the line given input of each product's common measured variable for the 7 products (the volume produced). Do I violate any assumptions of the binary logistic regression model by doing this?

Thank you so much for your time.