H O N C alternatives for organic life?
Its not scientifically or logically possible to say 'this is impossible' but since this is a speculative question, I'll hazard an opinion.  

I'd like to argue that, at Earth standard pressure temperature and atmospheric conditions, C H O and N are not really replaceable.  Throw water into the mix as the most common liquid and replacing any or all of these elements is hard to imagine.  

One can think about this by looking at the formation of long hydrocarbon chains.  Silicon can form polymers, but they tend to fall apart. Silcone-oxide polymers [SiR2-O-SiR2-O-] are more stable, but do not resemble hydrocarbon polymers as readily.  

Carbon is much more versatile, capable of forming either chain, or including sulfur, nitrogen in stable chains that don't seem to really have any length limitations. 

At thousands of atmosphere's pressure, temperatures of 70 Kelvin or in a sea of sulfuric acid, its easy to imagine other sorts of chemistry would dominate - phosphate , arsenic chains or germainium lattices, but on Earth most of these would fall apart.  

Even if/when life is discovered elsewhere , it will certainly feature chemistry that will completely amaze us; unlike anything here.  But even so, its hard for me to imagine it being completely exclusive of these four elements.  I just want to point out that by a wide margin, hydrogen is the most common element of the universe.   Nearly all the other light elements like C, O and N are the next most common.  This is simply because they take fewer nuclear fusion events to create.  Hydrogen is pretty useful no matter what chemistry you are looking at.  These four are probably going to be there somewhere - most planetary atmospheres have Ammonia, CO2 and methane in them as well - at least as far as we've seen.