Say I wish to do a linear model and my dependent variable $Y$ follows a binomial distribution: $Y_i \sim Bin(n_i,p_i), i = 1,...,N$. Usually I would use a generalized linear model and perform a logistic regression. But if the $n_i$'s are large enough, the $Y_i$'s will approximately follow a normal distribution, $Y_i \sim N(n_ip_i, \ n_ip_i(1-p_i))$. So in that case shouldn't I be able to simply perform a simple linear regression, without using a generalized linear model?
One of the assumptions of the linear regression is homoscedasticity (constant variance). In the binomial model, the variance obviously depends on $p_i$, which is the variable you're trying to model. As long as your curve is reasonably flat, or, in other words, the $p_i$'s don't vary too much, this will probably not be an issue. But still, if I were to review your results, I'd probably ask why you hadn't used logistic regression. What are the benefits of using a model which you know is less suited than another? ("All models are wrong, but some are useful.")