Generative models and machine learning promise accelerated material discovery in MOFs for CO2 capture and water harvesting but face significant challenges navigating vast chemical spaces while ensuring synthetizability. Here, we present MOFGen, a system of Agentic AI comprising interconnected agents: a large language model that proposes novel MOF compositions, a diffusion model that generates crystal structures, quantum mechanical agents that optimize and filter candidates, and synthetic-feasibility agents guided by expert rules and machine learning. Trained on all experimentally reported MOFs and computational databases, MOFGen generated hundreds of thousands of novel MOF structures and synthesizable organic linkers. Our methodology was validated through high-throughput experiments and the successful synthesis of five "AI-dreamt" MOFs, representing a major step toward automated synthesizable material discovery.