Like so many things, the answer to this question depends on what exactly you mean by it. There are certainly philosophical games you can play to show that eithe

Like so many things, the answer to this question depends on what exactly you mean by it. There are certainly philosophical games you can play to show that either answer is "possible": brains in a jar, malicious trickster deities, recordings on repeat, and the like. But if you're asking about what the best available scientific evidence supports, there is absolutely compelling evidence that this question must be answered in the framework of quantum mechanics. (There are experimental tests that can distinguish between quantum and classical predictions, and their results are unambiguous.) Within that framework, the most natural interpretation of this question might be, "Would it be possible in principle for a conscious being with perfect knowledge of the current state of the universe to predict the outcome that they would observe for any possible experiment?" Quantum mechanics answers this with an definitive "No." Even very simple systems can be prepared in a "superposition" of two possible states, in a sense that represents not just our ignorance of some underlying truth but a true mixture with measurable consequences. On the other hand, you might intend the question to mean "Given complete knowledge of the current state of the universe, would it in principle be possible to predict the future state of the universe?" If we leave aside issues of "observers" and "collapse of the wave function" for the moment, quantum mechanics clearly says that the answer is "Yes". (The mathematical property involved is "unitarity of time evolution".) If you begin with an electron in a superposition of "spin up" and "spin down" states and make a measurement, then a few moments later your experimental apparatus will be in a specific corresponding superposition of "detected spin up" and "detected spin down" states. But where you go from there depends on your preferred interpretation of quantum mechanics. The traditional Copenhagen interpretation says that once an observer looks at the measurement, the superposition state (the wave function) "collapses" to one answer or the other probabilistically, so determinism is lost. A "many-worlds" style interpretation would instead imply that the superposition of both outcomes persists in the state of the universe, but any particular instance of a given observer will see an outcome consistent with one measurement or the other (and the phenomenon of "decoherence" prevents those instances from interfering with each other in any meaningful way in the future). [Edit to add a disclaimer: There are weird open questions related to quantum gravity and black holes that call the unitarity of quantum mechanics into some doubt: a study of effects like Hawking radiation that arise when studying quantum fields in curved space-time might suggest that information about the state of the universe is lost over time. My impression as someone working in this field is that most people expect deep down that unitarity will be preserved in an ultimate theory, but there's no way to be sure until we have one. Also, LQG folks may be less confident of how their model would ultimately answer that question than string theory folks like me tend to be.]