We present limits on the molecular gas content of Lyman break galaxies (LBGs) at from observations targeting redshifted CO(1-0) and CO(2-1) line emission. We observed a single field containing eight spectroscopically confirmed LBGs, seven of which are contained within a narrow () redshift range and the eighth is at z = 5.2. No source was individually detected. Assuming the CO to H₂ conversion factor for vigorous starbursts, we place upper limits on the molecular gas content of individual LBGs of M(H₂) . From a stacking analysis combining all of the non-detections, the typical LBG has an H₂ mass limit comparable to their stellar mass, . This limit implies that, given the star formation rates of these systems (measured from their UV emission), star formation could be sustained for at most Myr, similar to the typical ages of their stellar populations. The lack of a substantially larger reservoir of cold gas argues against the LBGs being UV-luminous superstarbursts embedded in much larger UV-dark systems and as a result increases the likelihood that at least those LBGs with multiple components are starbursts triggered by mergers. The sources responsible for re-ionization are expected to be starbursts similar to these systems, but with lower luminosities, masses and consequently with star formation time-scales far shorter than the recombination time-scale. If so, the ionized bubbles expected in the IGM during the re-ionization era will only infrequently have UV-luminous sources at their centres.