I think the comments by StoneyB and the OP have it more correct than the existing answers...
It's not really fair to try to read this sentence without context.
For the first five years (1969–1974), the ratio of N to P in fertilizer was added at 12: 1 by weight, well above the Redfield ratio, to ensure that phytoplankton had adequate N and P supplies during the period when we were testing the C limitation hypothesis
The sentence you refer to involves data from 1969-1974, with adequate N and P.
I would rewrite the sentence as:
(Short-term bioassays) indicated (carbon-limited photosynthesis and algal growth), and (therefore) did not predict the continued growth of algal biomass in proportion to P (i.e., phosphorus).
Or, maybe in English that is more clear:
Algae were carbon-limited so the level of phosphorus didn't affect algal growth.
I think what probably happened is that in editing they went from:
Short-term bioassays indicated that carbon limited photosynthesis
(meaning their tests showed carbon was limiting photosynthesis)
Short-term bioassays indicated carbon-limited photosynthesis
(meaning their tests showed the lake was in a state called 'C-limited/carbon-limited photosynthesis'...which has the same overall meaning as above)
and left in the "that" when converting to the compound adjective.
Caveat: I'm a neuroscientist, and plants don't have brains, so this is typically a bit outside my wheelhouse... I'm giving this answer as someone familiar with science writing, not someone familiar with lake eutrophication except as a childhood neighbor to a eutrophic lake with a terrible phosphorus problem resulting from nearby farms.