If your intervals that you divide things up with do not overlap, I would actually suggest you *disregard* the interval censored aspect of your data, and merely treat it as ordinal/discrete. For non-parametric and semi-parametric interval censored data estimators, if the intervals do not overlap, your results are exactly equivalent to the results if you had treated them as discrete ordered outcomes (ie 1 = [0,1), 2 = [1,2), etc). As such, special software is really needed; you could easily use `R`'s `ordinal` package or even `coxme` for mixed effects models. 

If for some reason that doesn't currently make sense to me, your response intervals were not overlapping (ie for some reason you believe subject 1's exact time was in the interval [6-8), but you also believed subject 2's exact time was [7-9)) OR you're really committed to using fully parametric models, you can fit interval censored regression models (fully parametric AFT models can be found in the `survival` package, non-parametric, semi-parametric and fully parametric proportional odds and proportional hazards models can be found in my own `icenReg` package). 

But I'm not aware of any mixed effects for models interval censoring data at the moment. If you **really** wanted a parametric mixed effects model, you could hand code your model into something like `Stan` or `RJags` (my understanding is that they both have syntax that allows for interval censoring). But I would strongly suggest using the `ordinal` or `coxme` packages.