There are certain biologically-determined features of human beings that function as limitations on human happiness, regardless of the system of government that is in place. I wish to list a few of these, sticking to limitations whose existence few would deny and for which there is substantial evidence, without getting into speculative evolutionary biology territory. These are things that are unlikely to be remedied by any system of government or intervention. The only change that might remedy these limitations is a biological one - for instance, a physical change from a human body and human brain to a different sort of cognitive architecture.
I am using the term "happiness" in a broad sense here, not merely in a hedonistic sense, so that it includes things like satisfaction of preferences as determined by the individual (which preferences must, of course, be partially biologically determined).
Parents exercise control over children, despite the mismatch between parental interests and the interests of the child. A child cannot control himself, yet no one else's interests match his own. His parents' interests are frequently extremely different from his own - even a stranger would be more likely to make decisions on the child's behalf that reflect the child's values, rather than the parents. Of course, an unrelated person is also much more likely to abuse or kill the child.
Inequality is ubiquitous, overdetermined, and harmful. Inequality is ineradicable, both in terms of initial endowment (which is determined by necessary genetic variability) and in terms of economic distribution, regardless of the system of distribution. Inequality, in and of itself, causes suffering, regardless of the absolute level of material prosperity of the population. (The murder rate, for instance, more closely tracks inequality than overall poverty.) Even given an initially equal distribution, inequality would be guaranteed by the human preference for comparative well-being, rather than absolute well-being.
Humans' interest in maximizing sex is limited by the suffering caused by violent sexual jealousy. Neither the interest in sex nor the jealousy is likely to change.
Predominant heterosexuality, combined with the fact that the two genders experience different (average) sex drives, ensures frustration. Again, neither of these is likely to change. Widespread polygyny is not a solution that ensures a reduction in overall suffering, even where it is freely chosen by all partners, as it necessarily imposes celibacy and suffering on a large number of men who are not party to the polygynous arrangement.
Exploitation will always be lucrative, and sociopaths and other "cheaters" will always be with us. Sociopathy, as an evolutionary beneficial strategy at low frequencies, occurs (with low frequencies) in all human populations. Given any set of rules, from the ultraminimal state to totalitarian communism, figuring out how to successfully deviate from the rules in a way that benefits the individual will always be a benefit.
And, of course, there's this.
These, along with widely recognized limitations on human happiness such as aging, mortality, limited resources, and the lack of inherent meaning in life, act to limit the realistic expectations for happiness in even the most just nation imaginable. There may be hope for posthumans, but there is no realistic hope for humans to live lives of unlimited happiness. In many ways, suffering is guaranteed.
Edit: Here's another one.
Shame is necessary for human development and social functioning, and necessarily unpleasant. The presence of shame is ineradicable, nor would it be desirable to eradicate it. Nevertheless, it causes intense suffering for many, with the exception of those with antisocial personality disorder.