We have used medium-resolution spectra to search for evidence that proto-stellar objects accrete at high rates during their early 'assembly phase'. Models predict that depleted lithium and reduced luminosity in T-Tauri stars are key signatures of 'cold' high-rate accretion occurring early in a star's evolution. We found no evidence in 168 stars in NGC 2264 and the Orion nebula cluster for strong lithium depletion through analysis of veiling-corrected 6708{AA} lithium spectral line strengths. This suggests that 'cold' accretion at high rates (dM/dt>=5x10^-4^M_{sun}_/yr) occurs in the assembly phase of fewer than 0.5 percent of 0.3<=M*<=1.9M_{sun}_ stars. We also find that the dispersion in the strength of the 6708{AA} lithium line might imply an age spread that is similar in magnitude to the apparent age spread implied by the luminosity dispersion seen in colour-magnitude diagrams. Evidence for weak lithium depletion (<10% in equivalent width) that is correlated with luminosity is also apparent, but we are unable to determine whether age spreads or accretion at rates less than 5x10^-4^M_{sun}_/yr are responsible.