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First I describe the situation as I understood it. You have measurements (not assumed to have a normal or any other distribution) on $n$ individuals, six observations on each individual, on two different conditions $A,B$. We can write this as $$ y_{ijA}, y_{ijB} $$ for $i=1,2,\dotsc,n$, $j=1,2,3$. This could be modelled as an ANOVA with one random and one fixed factor, we can write a linear model like $$ y_{ijk} = \mu + \eta_i + \beta I(\text{$k=A$}) +\epsilon_{ijk} $$ This is one way of taking care of (that is, modeling) the dependence of the observations pertaining to the same individual. Here $\eta_i$ is a random effect for each individual and $\epsilon_{ijk}$ is the error term. (It might need some extra restrictions for identifiability). This could be estimated with standard software for linear mixed models, like lme4 in R. I don't know about nonparametric tests for such models ... but you could use bootstrapping, maybe, or bayesian methods. For references, look at any book about mixed models, if you are using R then maybe: Bates it is accessible and very good.