The systemic problem with currency unions, and this is true of all historical currency unions, not just the current Eurozone, is that the banking systems that comprise them expand their money supplies (money in this answer is the total sum of bank deposits) at different rates. Banking systems can also contract their money supplies, this is rarer, but equally problematic.
Between different currencies this issue is reflected in the valuation of currencies relative to each other, and over time this shifts more or less in relation to the different expansion rates, and to economic changes which can also influence the price level.
Within a currency however, this can't happen, and the pricing issues created by different expansion rates have to be resolved within the economy, in so much as they can be.
This problem can and does occur within individual countries. Different banks also expand at different rates within a banking system, and because banking is structured geographically, unbalanced flows occur everywhere. The London problem is good example of this in the UK.
However, within a country, it is usually possible to politically intervene and direct tax spending by the government to the regions that are expanding at slower rates. Germany does this very well, the US increasingly not so well, as banking becomes more concentrated there, and political pressure is being put on the US Government to reduce spending, and since China has been reduced to building empty cities in an attempt to re-direct funds, I wouldn't describe that as exactly healthy.
A large part of the problem is that this effect is very slow. Monetary expansion rates in fiscally stable countries are typically of the order of 1.3 - 3 times per decade. The stresses on the systems build up very slowly, are generally misunderstood - the symptoms are economic and so people's behaviour can be blamed. In that borrowing is the trigger for the expansion, behaviour also influences the problem (and generally amplifies it).
Whether or not breaking currencies down further would be a solution to this is an open research question. Back in the 18th century in the USA there was a time when this was effectively occurring as each bank issued its own notes, and these notes traded at different rates to each other. It can be done, whether or not we would want to live in that world is another matter entirely. The Eurozone was created for a number of reasons, and one was that having a single currency significantly simplified commercial transactions. The idea at the time was that countries who joined it would have sufficiently similar economies and be fiscally responsible with respect to debt and government spending. The intrinsic monetary expansion issues inherent in fractional reserve banking appear to have never been explicitly acknowledged.