Addenbrooke's Hospital has said a cap on Tier 2 visas is preventing it from employing the junior doctors it needs.
Senior figures at the Cambridge hospital say the system currently prevents them from hiring well trained medical staff with good english from overseas. Even in positions which can't otherwise be filled.
Three doctors from India, Pakistan and the Carribean who'd been offered jobs at the hospital recently had applications turned down by the Home Office.
It has increased the workload on other staff and means money has to be spent employing locums.
Fewer British people are currently training in medicine and with Brexit looming fewer are coming here from the EU.
The city's MP says capping the number of well qualified people who can come here is counterproductive.
The visa and associated costs can exceed a thousand pounds and cost even more if the applicant brings family.
ITV Anglia's Matthew Hudson reports: