Fully Homomorphic Encryption (FHE) allows arbitrarily complex computations on encrypted data without ever needing to decrypt it, thus enabling us to maintain data privacy on third-party systems. Unfortunately, sustaining deep computations with FHE requires a periodic noise reduction step known as bootstrapping. The cost of the bootstrapping operation is one of the primary barriers to the wide-spread adoption of FHE. In this paper, we present an in-depth architectural analysis of the bootstrapping step in FHE. First, we observe that secure implementations of bootstrapping exhibit a low arithmetic intensity (<1 Op/byte), require large caches (>100 MB), and are heavily bound by the main memory bandwidth. Consequently, we demonstrate that existing workloads observe marginal performance gains from the design of bespoke high-throughput arithmetic units tailored to FHE. Second, we propose several cache-friendly algorithmic optimizations that improve the throughput in FHE bootstrapping by enabling up to 3.2x higher arithmetic intensity and 4.6x lower memory bandwidth. Our optimizations apply to a wide range of structurally similar computations such as private evaluation and training of machine learning models. Finally, we incorporate these optimizations into an architectural tool which, given a cache size, memory subsystem, the number of functional units and a desired security level, selects optimal cryptosystem parameters to maximize the bootstrapping throughput. Our optimized bootstrapping implementation represents a best-case scenario for compute acceleration of FHE. We show that despite these optimizations, bootstrapping continues to be bottlenecked by main memory bandwidth. We propose new research directions to address the underlying memory bottleneck. In summary, our answer to the titular question is: yes, but only after addressing the memory bottleneck!