Aberration-corrected microscopes with subatomic resolution will impact broad areas of science and technology. However, the experimentally observed lifetime of the corrected state is just a few minutes. Here we show that the corrected state is intrinsically unstable; the higher its quality, the more unstable it is. Analyzing the contrast transfer function near optimum correction, we define an "instability budget" which allows a rational trade-off between resolution and stability. Unless control systems are developed to overcome these challenges, intrinsic instability poses a fundamental limit to the resolution practically achievable in the electron microscope.