Is it still possible for a mathematician to be competent in most of the areas or fields of mathematics?
No, it is no longer possible, and has been impossible for about 100 years. Hilbert and Poincaré are considered to be the last mathematicians to have mastered so much of the mathematics of their day that they could create and build in most fields.
Mathematics has grown in both breadth and depth to such an extent that producing original research in more than a few areas is extremely difficult, and writing papers cutting through "all" modern fields of math is truly impossible.
As an example, Fields medalist Tim Gowers is well known to have done groundbreaking work in several distinct fields: the theory of Banach spaces and combinatorial number theory, to roughly portray the two main ones. This is astonishing, but it also helps establish the limits of what even the best mathematicians today can be expected to achieve. Gowers does not work in model theory, algebraic geometry, representation theory, partial differential equations, group theory or any number of other domains that are simply outside of his fields of expertise.
In 2010, while attending the International Congress of Mathematics, Gowers wrote:
...I decided on two prodigies: Marianna Csörnyei and Jacob Lurie. Marianna I have known quite well for many years, and she works in areas that I can be expected to understand reasonably well (which is not quite the same as saying that I do understand them reasonably well, but in fact I do usually follow quite a bit of her talks). Jacob Lurie is the opposite: I had never met him, or even seen him, and had absolutely no chance of understanding anything he would say, so I was going for the sole purpose of gawping.
Here is one of the most prominent mathematicians of our time admitting freely that the work of a colleague is completely beyond his reach. You have to admire the honesty (and the wonderfully understated humor), but frankly this isn't surprising: the frontiers of research in higher category theory (Lurie's field), or logic, or arithmetic geometry or pretty much anything nowadays are almost completely obscure to anyone but the experts in those domains.
None of this is peculiar to Gowers, of course. The same can be said of any Fields medalist in the last 70 years, or any prominent mathematician. Even the most diverse ones like Serre, Gel'fand, Connes, Manin, or more recently Tao all have clear domains of focus, albeit breathtakingly wide and deep ones.
Go to a colloquium talk at any math department and you'll see that almost all of the talks have two parts: a simplified, high-level, almost metaphorical survey of the gist of the topic, and then an excursion into the actual details of a proof which is either still very high level or else it is clearly incomprehensible to anyone except two faculty members and one grad student, the one whose advisor is the speaker herself.
No joke. This is how things are, which tells you that most mathematicians can't even fully understand the works of others outside their domain, let alone produce papers in such different domains.
This isn't a great state of affairs but, until we find ways to dramatically improve the abilities of our minds or the length of our learning-worthy life spans, I suspect this is how it will remain.