How are superconductors discovered? How do scientists discover superconductors? Do they test properties of every material available on Earth? Or do they do something mathematically?
 A: One of the biggest failure of theoretical condensed matter and/or material sciences is that up to now, nobody has ever been able to predict what compounds will be a good superconductors.
Of course, since we don't really understand High-Tc superconductivity, we cannot predict which ceramic will or will not be a nice superconductor. But even in the case of more standard superconductors described by BCS or more refined theories (like Eliashberg's theory), the predictive power of theoretical approaches is close to zero. To summarize, all superconductors are found experimentally, and then theorists try to explain why this particular alloy/compound has these properties.
A: "Testing every material" has been tried with combinatorial methods- ink jet printing of test compositions, fire, look.  Nothing interesting.  As with ceramic supercons, the answer is not to be found under a bright streetlight of prior success.  It sits in the dark middle of the block where it offends goodthought.
BCS supercons cannot be predicted, but folks thought they had a handle on what lattice structures were fertile and how high the maximum Tc could be re phonon softening.  They were not quite right (wrong),
http://cmp.physics.iastate.edu/canfield/pub/pt0303.pdf
Hight-temp ceramic supercon structures look like smectite clays - terrible for fabrication.  How they operate and how they can be further improved are stalled.  William A. Little, Phys. Rev. 134 A1416-A1424 (1964). Exciton-based ambient temperature superconductors: polyacetylenes  substituted with polarizable chromophores, [-C(Ar)=(Ar)C-]n or [=(Ar)C-C(Ar)=]n (same thing, just shifted a bond).  No  insulating links are allowed between the polyacetylene core and the -Ar chromophores.
Little suggested replacing BCS large mass phonons (quantized lattice vibrations characterized by Debye temperature) with small mass excitons (quantized electronic excitations) possessing characteristic energies around 2 eV or 23,000 K.  Exciton-mediated electron pairing suggests superconductor critical temperatures substantially exceeding 300 K even with weak coupling.  The polymers were absolutely 1960s unmakable.  Proposed Tc were enthusiastically ridiculed.  In the 21st century, Little's polymers are trivially synthesized on paper:  
aryl aldehyde to diaryl benzoin to diaryl benzil, + Tebbe methylenation to 2,3-diarylbutadiene 
2,3-diarylbutadiene + Grubbs or Schrock catalysts, ADMET polymerization to 
Little  + ethylene
ArCHO to ArC(=O)-CH(OH)-Ar to O=C(Ar)-(Ar)C=O to H2C=C(Ar)-(Ar)C=CH2 
 H2C=C(Ar)-(Ar)C=CH2 to 
[=C(Ar)-(Ar)C=]n  + H2C=CH2  
One can diddle candidate Little exciton supercons in software, Hyperchem Lite.  Look at that pi-stacking!  One would add long chains to each mer to solubilize the polymer.  Hydrogens and pi-bonds have been omitted for ease of stereo-viewing,
http://www.mazepath.com/uncleal/pave1.png 
  (Amine rather than ether linkages are synthetically easier)
There is a procedural problem.  Supercon research is physical theory and engineering.  Organic chemistry is obviously irrelevant to both.  You cannot discover that which you postulate cannot exist, except as an act of insubordination.  That might be why they fail.
