Do materials gain compressive or tensile strength if compressed? If I take a block of play-dough and squish it down, it becomes harder to pull apart and push together.
Does this same phenomena hold for "real world" materials? If I take a block of steel and compress it, will it become stronger in tension or compression - and if so, by how much?
 A: To a first approximation, we can generally assume that the strength and yielding behavior of engineering materials are independent of pressure. In fact, the widely used Tresca and von Mises yield criteria, for example, completely ignore pressure (also known as hydrostatic stress, also known as equitrixial compression). This aspect can be seen in the "cylinder of safety" that lies around $\sigma_1=\sigma_2=\sigma_3$ (i.e., the hydrostatic axis); the material is not predicted to fail unless the stress state breaches the cylinder's surface:

Experimentally, however, we do find an increase in strength with increasing pressure at a rate of approximately 1/15. Thomas presented the following data for four types of steel: 
Thomas, T. Y. "Theoretical effect of large hydrostatic pressures on the tensile strength of materials." PNAS 57.5 (1967): 1195.
and Spitzig et al. presented related data for tension and compression of 4310 steel:
Spitzig, William A., Robert J. Sober, and Owen Richmond. "Pressure dependence of yielding and associated volume expansion in tempered martensite." Acta Metallurgica 23.7 (1975): 885-893.
Spitzig and Richmond parameterized the strength (as expressed by the flow stress $\sigma$) as a function of the gauge pressure $p$ as $$\sigma=\sigma_0(1+3\alpha p)$$ where $\sigma_0$ is the atmospheric yield strength and $\alpha$ is a constant given for different materials in the following table (Spitzig, W. A., and O. Richmond. "The effect of pressure on the flow stress of metals." Acta Metallurgica 32.3 (1984): 457-463) (red rectangle mine):

Increasing pressure also results in increased ductility, notably so for materials that are brittle at atmospheric pressure (i.e., samples that would normally snap under tension tend to neck).
