Could be the high and low limit screws are set such that they do not allow the rear derailleur to shift to those gears. It could also be that they chain is too short, but I would expect that it could shift to the smallest cog in rear cassette when on the small chainring since in the position there is still slack in the chain.
What are you shifting with? Downtube? Bar end? Brifter? I suspect a mismatch between shifter/derailleur/cassette. Maybe the limit screws, but that seems unlikely. It's not the chain, though follow the advice to get correct chain length. Too short a chain will sooner or later put the derailleur into your spokes.