Models for binary response that appear like logit/probit regression but never approach 1 I'm reading the textbook "Introduction to Econometrics" by Stock and Watson. Chapter 11 discusses regression with a binary response and it teaches the logit/probit models mostly. These seem to make sense when the plots show an S shaped relationship between X and Y, but what if I'm working with a dataset where the model calculates a probability in the shape of a "hill".
For example what if my continuous predictor X ranges from 0-100 and my response Y is only equal to 1 for X's at values of 30-70 but gets more concentrated at X's around 50. So lets say around X=50 we could have a 40% probability of Y=1, and then at X=40, X=60 we'd have 25% probability of Y=1 until the probability of Y=1 dissipates around 30-70. I cant find any models that model such a relationship through google - Are there any models that are more appropriate than logit/probit for this type of relationship?

Logistic regression is a fine approach even in this case.  The problem is that a simple linear relationship (on the log odds scale) is insufficient to model this relationship.  You need to introduce non-linearity into the conditional mean, and this is most easily done with a spline.
Rather than let
$$\operatorname{logit}(p) = \beta_0 + \beta_1 x $$
let the linear predictor be
$$\operatorname{logit}(p) = \beta_0 + f(x)$$
Where $f(x)$ is some spline function.  Because splines can be represented as linear combinations of non-linear functions, we can estimate this using the same machinery we do for linear models.
Shown below is an example.  In red is the true conditional mean, which was used to generate the black dots.  In black is the estimated conditional mean using a natural spline.

