Robots can do this too, there's no real time benefit (but a robot can do more than 10 peptide couplings a day and doesn't need a lunch break -- though I guess neither does a grad student).

Chief problems include: you need super concentrated protected peptides, which are highly insoluble in just about any solvent, so your system needs to have tons of filters, plus stuff gunks up your apparatus over time anyways. Then, fmoc craps out at about 20 amino acids if you're lucky, due to exponential decay of yields (99%^20, not to mention later yields can be lower due to peptide chains curling on themselves), boc can go longer. Not to mention separating what you want away from side products gets difficult.

Finally the wastefulness: boc chemistry you're throwing away atoms almost 1:1 with what you're putting in to the peptide. Fmoc is typically 3:1. The solvents you use are these nasty greasy fishy smelling solvents (dmf, nmp). And the activating agents are really expensive compounds that you will slowly become allergic to if you keep working with peptides.

I personally have, though unpublished work, the first total (artificial) synthesis of ubiquitin, that I did as a grad student in my first year. Although it's 76 amino acids we synthesized it in two parts and stitched them together using a technique called "native chemical ligation", though we did something special called "folding-assisted native chemical ligation".