I've got an oil fired combi (Worcester Heatslave) that is sixteen years old and still ok. I have replaced jets, burner motor, diverter valve motor, baffle plates etc but it is still fine. I am told that is the main water tanks splits then that pretty much writes it off as it is an uneconomical repair. I'd have to weigh up doing part of that repair myself before deciding to bin the whole thing.
This model stores about 45litres of water so in some ways it isn't quite a true combi - I think this is the case for most oil fired versions though. It will easily supply a very hot shower at 25litres/minute at 3 bar which is a lot. I'm actually waiting for some restrictors to arrive to throttle the showers down to more like 10litres/min as they're a bit OTT at the moment.
For me, one of the downsides to a combi is the need/desire to over spec them slightly when compared to the true heating load of the house. This seems to be common practice to get the hot water supply to be ample. The trade off is short firing cycles for the boiler as the heating output is a little too high - a long steady burn is a lot more efficient and this is one advantage of a proper cylinder based system.
Another disadvantage is that it is hard to integrate solar hot water and other heat sources like wood gasification or back boilers to a combi system. A tank makes it easy as you have a natural neutral point to connect everything to.
I'm actually in the process of speccing up a complete new system with a 15kW gasification boiler stove, and an array of solar tubes to reduce my reliance on oil. It's a lot of money to spend up front so I'm working out what sort of payback periods I'm looking at for various options.