Does a high likelihood of duplicate samples invalidate my data or certain operations on it? We have data from random polling within a group of people, sampled over a long period of time. As the pool is relatively small and anonymity is crucial there is a high likelihood that we have duplicate samples (though we can't tell for certain because we'd expect responses to vary over time).
Does this somehow invalidate the data, or mean that certain operations on it will not be meaningful? Or is it OK to proceed as normal so long as I state this likelihood?

I'm happy to provide more detail if needed - just say what would help in a comment.
Also I'm making this CW. Please feel free to edit the question if there are other relevant implications of duplicate data that would be worth specifying.
 A: You would normally make the assumption of independence of observations in your modelling.  
Alternatively if you expected correlation between observations it would be good to model this and estimate that correlation. You can't do this as you don't know which observations are likely to be correlated.  
If you assume independence when some observations are in fact positively correlated you will understimate the between subject variance. This means you are more likely to find "significant differences" than statistical theory would suggest. You can think of it as appearing to have more samples than you in fact do have as some are almost repeats.
A: You should adjust your standard errors (and p-values, confidence intervals, etc) to account for the observations not being independent. You can do this under some reasonable assumptions even though you don't know which observations are of the same person. 
For example, suppose you're estimating the mean of some variable, x. Let x_{it} be the observations in your sample with the understanding that x_{it} and x_{it+1} are not necessarily the same person. Let e_{it} = x_{it} - mean(x). A central limit theorem for dependent data (e.g. this one) tells us that the sample mean is asymptotically normal with variance
V = lim E[1/NT (sum e_{it})^2 ] = lim 1/NT sum E[e_{it}e_{js}]
where the limit is as NT -> infinity, the first sum is over i,t and the second is over i,t,j,s. Assume that e_{it} are uncorrelated for different individuals. Suppose for a fixed individual, that e_{it} is weakly stationary with v_s = E[e_{t} e_{t+s} ]. Also, let's suppose that probability of the same person being sampled at time period s conditional on being in at time t is p. If the pool and sample sizes is fixed, then p = (sample size)/(pool size). Given that, we know
E[e_{it}e_{js}] = v_0 if t=s else p v_{t-s}
Now assuming that T -> infinity, we get
V = v_0 + p 2 sum_{t=1}^infinity v_t
This is the asymptotic variance of the sample mean. To use this result, you need to be able to estimate V consistently. V is called the long run variance. If N is fixed, it can be estimated by kernel methods. If N -> infinity seems like a more appropriate asymptotic approximation, then you need not resort to kernels. 
All this is just meant as example to get you started. You can modify many of the assumptions if they're not appropriate for your setting. For example, p could vary with t and s instead of being constant. If you're not just interested in sample averages, you can do similar calculations for any statistic you want. 
