Sure I can. I can call up my family doc and book an appointment (generally within 1 week), or if I don't want to wait that long I can go to the local clinic and wait for the on-call doc to see me, or if it's really serious I can go to Emergency at the hospital.
Now if you want to see a specialist then you might be waiting a while. But if the GP thinks your case is serious then they'll bump you further up the queue, and if it's really serious they can generally get you in right away.
The biggest failing is in highly specialized stuff like pediatric psychiatry, or health care for really remote areas, but I suspect that would be an issue under most forms of health care.