As @eclaassen pointed out, there are methods/tools to do survival analysis with random effects. You might be interested in this SUGI paper: Generalized Linear Mixed Model Approach to Time-to-Event Data with Censored Observations . But the JMP does not offer such tools in the current release. Please pay attention to her new platform in the coming release.
Besides GLIMMIX type tools, Bayesian inference might be a different approach. I am working on a new Repeated Measure Degradation platform, which models random subject parameters using Bayesian inference. The platform itself is not relevant to what you need now, though the technology can be applied.
For you rule-of-thumb question, a major concern is how censoring affects your objective of the analysis. There are two things about censoring that you may need to worry about: how many censored observations occur, and when (or where) censored observations occur. A worrisome situation might be that you have a right censored data with large amount of censored observation, while you are interested in failure probability far beyond the censoring time. But it is not worrisome, if your interest is in the failure probability before censoring time, and you still have a large amount of failure observations, even though censored proportion is big.
In the case that censoring has a great impact on your results so the results have great uncertainties, one needs to put very strong assumptions, either about a specific model, or about one or more parameters.
A classic method known as Weibayes provides a useful approach when almost all observations are censored. In addition to that method, Bayesian inference is another and more comprehensive approach to handle heavy censored data. JMP Life Distribution and Fit Life by X platforms provide these methods accordingly.
