The traditional view that Ly-alpha emission and dust should be mutually exclusive has been questioned more and more often, notably the observations of Ly-alpha emission from ULIRGs seems to counter this view. In this paper we seek to address the reverse question: How large a fraction of Ly-alpha selected galaxies are ULIRGs? Using two samples of 24/25 Ly-alpha emitting galaxies at z = 0.3/2.3 we perform this test, also including results at z = 3.1, and find that whereas the ULIRG fraction at z = 3.1 is very small, it systematically increases towards lower redshifts. There is a hint that this evolution may be quite sudden and that it happens around a redshift of z ~ 2.5. Measuring the infrared luminosities of the Ly-alpha emitters, we find that they are in the normal to ULIRG range in the lower redshift sample, whereas the higher redshift galaxies all have luminosities in the ULIRG category. The Ly-alpha escape fractions for these infrared bright galaxies are in the range 1-100 % in the low redshift galaxies, but are very low, 0.4 % on average, in the high redshift galaxies. The unobscured star formation rates are very high, ranging from 500 to more than 5000 Msun / yr. The dust attenuation derived are in the range 0.0 < A_V < 3.5.