Certainly the "above" part gives me the option to cross higher if I choose to, but that's not the point here. If I'm trying to get down, I could choose to descend to the lowest altitude allowed, which is 7800 in the example. Are you saying that I would need to plan my descent to hit my altitude (or above) right at the fix? In other words, I can't reach 7800 before I get to the fix?
Basically "cross <fix> at or above 7800" means this: I don't care what you do, or when you start your descent; you need to cross <fix> at 7800ft or above. If you immediately descend, that's okay. If you take your time, that's okay. To be honest, you could cross that fix at FL210, and would still meet the requirement for what you were told to do.
What it doesn't mean is that you are immediately cleared to descend to that altitude. Take, for example, "reduce your speed to 230kts, then cross <fix> at or above 7800."
You wouldn't be immediately able to descend (which would basically be at pilot's discretion) until the speed restriction is met.
Another question long these lines, what would be the difference between the clearance given, and "descend at pilot's discretion to 7800"? They seem to mean the same thing here - I own all the altitude between where I am and 7800.
Are you on a STAR? If so, what "cross <fix> at/maintain" would be primarily used for is to meet a crossing restriction as depicted on a STAR. Take for example, the KEPEC6 or the CLARR3 arrivals into KLAS:
http://155.178.201.160/d-tpp/1801/00662KEPEC.PDFhttp://155.178.201.160/d-tpp/1801/00662CLARR.PDFBoth have the 13000ft and 250K crossing restriction at CLARR. And seeing that you'd be joining the STAR anywhere from 60 to 120nm away, you definitely would not want to be immediately descending to 13000, especially if it may take you below the MVA/MOA for the segment in question.
If not on a STAR and receiving vectors, "cross <fix> at/maintain" wouldn't be a good call to give, because you'd either have to be properly equipped to know where <fix> is, or check if it is a non-RNAV fix, meaning that it should be defined by a radial from a given VOR and DME from that VOR. Regardless, you "should" be on a published chart to be given that call to cross a given fix.
BL.