A substantial proportion of the population has private health insurance in Portugal, where I work as a GP. Funding cuts and raised co-payments because of the financial crisis have decreased the appeal of public healthcare, prompting many people to buy private insurance.  This grants quicker access, and sometimes with smaller co-payments, than in the public system.
Many patients use private medicine for most of their needs and go straight to a specialist, but sometimes they have to see a GP, for example, to request sick leave or for a specific referral within the public healthcare system. Private GPs, which also exist, can’t do either. Many women may prefer a private gynaecologist or obstetrician for routine care and pregnancy follow-up, for instance, but they may want to give birth, particularly in cases of high-risk pregnancies, at a public hospital if they perceive the neonatal care to be better there.
Some patients work for big companies that have their own medical staff. But for reasons of confidentiality, these patients may prefer to disclose sensitive information to doctors outside of their work. Many richer patients may first see a GP in their 60s because they have had private insurance all their lives but find it harder to get cover in older age.
There are many legitimate reasons for patients with private insurance also to use the public healthcare system. But this demand, which is not necessarily urgent, may well make it harder for patients who lack insurance and who rely entirely on the public healthcare system to access healthcare. This is an example of the “inverse care law”: the people in greatest need of medical care often have most difficulties accessing it. 
Bureaucracy associated with assessing patients’ eligibility for care and administration of insurance in primary care, including collecting payment, come with costs, and they can cause frustrating delays and wastage in clinical time, as I’ve described before. 
1 Head to Head: Could private top-up insurance help fund the NHS?
5 Hart, JT. The inverse care law. Lancet. 1971 Feb 27;1(7696):405-12. N
Tiago Villanueva is assistant editor, The BMJ, and a locum GP in the Portuguese national health service.