Getting $>1$ responses in logistic regression? Is it normal to get values for response that are $> 1$ even though in logistic regression the response has meaning only in the range $[0,1]$?
Does one then have to truncate all $> 1$ values to mean just $1.0$?

The reason for asking is that
I got something that looked like it might have response $> 1$ using a skin variable (1 means skin that's more prone to get sunburn and 0 means normal skin), a trt variable (1 signifies that the patient took beta carotene and 0 that the patient didn't) and then a cancer variable that signifies, whether the patient got skin cancer during the study. So $cancer \text{ ~ } trt+skin$ produces $$cancer=0.1331410+0.5570949⋅trt+0.627890⋅skin$$
(these coefficients are after taking invlogit() of the coefficients returned by glm()) which is $>1$ if both variables $==1$. 
Although, when seeing whether such rows actually exist (where $skin=trt=1$), no such rows exist.
 A: You are not getting probabilities because you are confusing the order of operation between adding the coefficients and taking the inverse logit.
Explanation:
You have a logistic model given by
$$logit(Pr(cancer=1|trt,skin,\beta)) = \beta_0 + \beta_1 * trt + \beta_2 * skin$$
You are trying to calculate a probability (I denote this quantity by $Pr^*$, since it's not a probability) by doing the following:
$$Pr^*(cancer=1|trt,skin,\beta) = logit^{-1}(\beta_0) + logit^{-1}(\beta_1 * trt) + logit^{-1}(\beta_2 * skin)$$
When it should be this:
$$Pr(cancer=1|trt,skin,\beta) = logit^{-1}(\beta_0 + \beta_1 * trt + \beta_2 * skin)$$
Note that, in general
$$logit^{-1}(A+B) \neq logit^{-1}(A) + logit^{-1}(B)$$
We would say that a non-linear transformation on a sum of two variables is not equal to the sum of the transformed variables, except in special, trivial cases.
This would be easier to see if you used a log-linear model instead of a logistic model:
$$exp(A+B)\neq exp(A) + exp(B)$$
Since we know that 
$$exp(A+B)= exp(A)*exp(B)$$
In other words, to get a probability, you first find the log-odds at trt=skin=1 as
$logit(Pr(cancer=1|trt,skin,\beta)) = -1.873468 + 0.229380*trt + 0.5231755*skin$
$logit(Pr(cancer=1|trt=1,skin=1,\beta)) = -1.873468 + 0.229380*1 + 0.5231755*1$
$logit(Pr(cancer=1|trt=1,skin=1,\beta)) = -1.120912$
And then transform:
$$Pr(cancer=1|trt=1,skin=1,\beta) = logit^{-1}(-1.120912) = 0.2458422$$
