Electronic Health Records (EHRs) have improved many aspects of healthcare and allowed for easier patient management for medical providers. Blockchains have been proposed as a promising solution for supporting Electronic Health Records (EHRs), but have also been linked to scalability concerns about supporting real-world HealthCare systems. This paper quantifies the scalability issues and bottlenecks related to current blockchains and puts into perspective the limitations blockchains have with supporting HealthCare systems. Particularly we show that well known blockchains such as Bitcoin and Ethereum cannot support transactions of a large scale hospital system such as the University of Kentucky HealthCare system and leave over 7.5M unsealed transactions per day. We then discuss how bottlenecks of blockchains can be relieved with sidechains, enabling well-known blockchains to support large hospital systems of over 30M transactions per day. We then introduce the Patient-Healthchain architecture to provide future direction on how scaling blockchains for EHR systems with sidechains can be achieved.