If each variable enters in linearly, then 5 variables produces a model with only 6 unknown coefficients. polyfitn will have no trouble in the estimation, nor will you.
However, if you will allow higher order terms in the multi-nomial model, and fully general multi-way interactions, then you get a LOT of terms. And many of those terms are not very important in the model. So people use techniques to reduce a model, tossing away terms that seem to have no significance. There are additional problems however. Is your data sufficient to estimate all of those terms? In fact, it is easy to pose data and a model where you simply cannot form estimates of some of those coefficients, regardless of how much you want to do so. This gets into large fields of study such as design of experiments.
The model you seem to be initially posing looks to be a simple linear one, with no interaction terms at all. But even there, I cannot know if your data is sufficient to estimate the parameters you want to use. You are also asking questions that seem like you might want to use some stepwise regression tools, yet you lack the stats toolbox.
I would recommend that you first do a little reading about regression analysis, but of course a text like Draper and Smith (of moderate size long ago when I learned the topic) has ballooned over many editions. So you might well want to speak to someone with expertise in statistics and modeling. (Sorry, but it sounds like you are trying to do something that will require expertise, yet you have absolutely no expertise in the matter, nor do you have any tools that would prove to be of value in this.)
