There's several ambiguities in your question that an answer needs to address. First, I am going to assume that your uses of "should" refer to moral rather than legal determinations. Otherwise, this isn't going to fit under philosophy. Second, I'm going to answer in terms of applied ethics rather than formal ethical theories. Third, I will following your question, use rights language.
Assuming citizens have rights, there are several different ways to conceive of rights. There's Isaiah Berlin's positive and negative rights distinction. On this account, negative rights where the government should not interfere with a course of action. Positive rights reflect those cases where the government or some other party is compelled to act for you. "Keep the government out of the bedroom" articulates a negative right --> to be able to sleep with whom you choose when you choose. "The government should make sure no one goes hungry" is a positive rights claim that the government has a duty to supply everyone with food.
A second distinction some draw is between first-order and second-order rights. A first-order right is something you have a direct right to. A second-order right is something that you have in light of your first order rights. So if you have a first order right to live, then you may claim a second order right that the government give you food. If you have a first order right to speech, then you have a second order right to the means of speaking. (i.e., the government cannot prohibit you from making speeches in public areas if you have a right of free speech). Actually, some take this further and speak of 3rd order rights -- but that's irrelevant for our purposes.
We need one further distinction to complete the philosophical picture. There are absolute and mitigable rights. Absolute rights are those which are of "incomparable worth" (to use a construction from Kant) whereas mitigable rights are those of "price" -- meaning they are tradeable for other things of moral value.
Given these three axes, we can give some answers.
- For an absolute negative 1st order right, the government is always wrong to inhibit this right.
- For any mitigable right (negative or positive / 1st or 2nd order), the government could for a sufficiently good reason abrogate its responsibility to its citizens.
After that it gets a little cloudy mostly because the philosophical pieces break down.
For an absolute positive 2nd order right, it seem possible that some conflict could prevent its execution without being immoral. imagine for instance, a right to paper and printing to build on a right to free expression. Conceivably if the government has no paper, it has lost its ability to perform this right and therewith failed in this right -- but not necessarily morally. (or to put it another way, is it possible for a right to be 2nd order and absolute? Also can there be absolute 2nd order rights? i.e. rights where the government's duty of non-interference is not immediate but is absolute? )
For an absolute positive 1st order right, it would be immoral for the government not to supply but it's difficult to understand what would qualify as such a right.