Old-fashioned economic models (multiplier-accelerator models of the business cycle, for example) operate in historical time: outcomes in one period determine decisions in the next period. That is, agents are backward-looking. The Lucas critique is that this assumes that people cannot predict what will happen in the future. The analyst on the other hand can derive later outcomes from earlier ones (or we would not be able to tell a causal story), so why can't the agents in the model?
Lucas says this is an unacceptable contradiction, and resolves it by attributing to the agents the model used by the analyst. (Interestingly some Post Keynesians (e.g. Shackle) seem to see the same contradiction but they resolve it the other way, and take the inability to predict the future attributed to the agents in the model as a fundamental feature of the universe, so applicable to the analyst too.) But is the idea of predictable but unpredicted outcomes such an unacceptable contradiction?
One reason to say no is that the idea that agents must know as much as analyst rests on a sociological foundation – that institutions are such as to foster knowledge of the best estimate of future outcomes. This need not be the case. For example, consider the owners of an asset that has recently appreciated in value, where there is some doubt about whether the appreciation is transitory or permanent, or whether further appreciation should be expected. Those asset-owners who have a convincing story of why further appreciation is likely will be most successful at selling at a higher price, and so will increase their weight in the market. And to have a convincing story you should yourself be convinced by it – this is true both logically and psychologically. Similarly with various arm's-length relationships that must be periodically renewed – the most accurate promises are not necessarily the most likely to bring success. Or on the other side, classes and organizations to maintain their coherence need their members to hold certain beliefs. This could take the deep form of ideology of various kinds, or the simple form of the practical requirements of organizational decision-making implying a limited set of inputs.
The other reason comes if you carry the Lucas critique through to its logical conclusion. Those who accept rational expectations also use the method of comparative statics, where transitions from one equilibrium to another is the result of “shocks”. One set of technologies, tastes, endowments, policies, etc. yields equilibrium A. Then a shock changes those parameters, and now there's equilibrium B. Joan Robinson objected to this procedure on grounds that it ignored dynamics of transition from A to B, but there is another problem. Evidently B is a possible future of A. The analyst knows this. So why don't the inhabitants of A? Unless the shock is literally divine intervention, presumably its probability can be affected by the their actions, so doesn't the analysis of A have to take that into account? Or, even if the shock is indeed an act of God, it's possibility must be known – since it is known to the analyst – and so it must affect decisions made in A. But in that case the effects of the shock can be hedged and nothing happens as a result of the shock; there is no longer two equilibria, just one. So we either have agents with perfect knowledge of everything and no knowledge of shocks, which must literally be divine interventions; or we can have only a single equilibrium which nothing can change; or we can become nihilists like Shackle; or we can reject the Lucas critique and accept that there are regularities in economic behavior that are not anticipated by the actors involved.