DBAs are often «cost-centres» and self-directed time is often neither understood nor appreciated by those above. It can be hard when one has to account for one's time in 15-minute intervals.
Often one needs to explain what one does in terms of tangible benefits, 'I was working on improving the poor performance of the person search' and the like.
Often those above do not value 'prevention being better than cure' or preparation of technologies as yet unused in the company. You will not spend time learning about replication until we tell you that we need replication fully working within a month.
That being said, I incorporate self-directed time into my tasks. For example, I will wait the appropriate task that is sent my way to hone my skills with the WITH-clause and recursion. I could surely do the task faster the way I had been doing it before but I am thinking medium to long term. With the next such task I can solve the problem faster. I have a list of aspects of SQL Server that is on my 'to be practiced'-list.
It gets harder with something like a cube. One can't just learn all about data-marts, OLAP-types, MDX and DMX in the course of a query that might normally take 2 days to create and test. One would have to build it in one's spare time, populate it with company data, demonstrate it usefully to the right people and then when asked how long it would take to build, repay yourself with the expended time. It's not nice, it's risky but sometimes it's an effective way to introduce whole new features to those that pay.