The Just-in-Time manufacturing system (that heavily influenced Lean thinking) relied on small batch sizes of inventory between processing steps. In twentieth century mass production, it was believed that large amounts of inventory were necessary - even efficient - for a properly running factory. Taici Ohno at Toyota challenged this idea by regularly reducing inventories. What would happen was that problems would surface. But these were the problems that were they the whole time. The large inventories were only a symptom, covering the real issues from sight.
Take away: Challenge the assumption that you are absolutely can't live without something. Once you start reducing it, pay attention to what happens next. These are clues for the things that could really be holding you back. Now that you know that, what are you going to do about it?