"Already" refers to some condition C that has begun in the past and is still in effect, in practical terms or with regard to its impact on the speaker:
"Still", on the other hand, is relative to a condition currently ongoing but bound to finish at some point in the future:
So, it would be odd to qualify a given event with both adverbs in the same sentence,
because each would claim the statement focus: one is hard-pressed to find real-life examples of this configuration.
This evening I went out for dinner at around 9:00PM or so. As daylight saving time has just kicked in in Spain, yesterday was the first time in the year that the setting sun can still be seen hiding behind rooftops at such a late hour. I exclaimed:
¡Ya es todavía de día!
which could be roughly translated to:
There's already still daylight!
So there you have it, "already" and "still" in the same sentence. In reality, the temporal structure of the sentence is not like depicted above, but rather the following:
C is the moment daylight time saving goes into effect, and C' the situation where the sun is out, that is, between dawn and dusk.The funny thing about the statement is that "already" refers to C and "still" to C', a sort of multi-scale sentence. A clearer paraphrasing reveals this:
The day has already come when there's still daylight this late!
I very clearly remember one early spring evening when I was around nine years old: those days I was taking extra-class music lessons and came back home late, in total darkness, but that very Monday when I approached the block the sun was still pouring an afterglow of dusty golden light over the horizon line. I think this was the first time I realized days shrink and grow during the year; I guess that Monday was the first one in daylight saving time (which I knew nothing about back then) and shifting had just happened the day before, so making the effect of ever expanding sunlight much more noticeable.