If bortezomib, a cancer drug, inhibits cell proteasomes, wouldn't resulting protein aggregate in normal cells further increase the risk of cancer?
Proteasome inhibitors do affect normal cells to some extent, but the whole point of using them as cancer treatments is that (some) cancer cells are far more sensitive to proteasome inhibitors than are normal cells.  For example, multiple myeloma cells (the first clinical targets of Bortezomib) that overproduce immunoglobulins are dependent on proteasome degradation to eliminate them and are therefore sensitive to Bortezomib.  There are many other aspects of cancer cells that may make them more dependent on proteasome; see for instance 

Proteasome inhibitors in cancer therapy: Treatment regimen and peripheral neuropathy as a side effect for some examples.

But of course there are effects on normal cells as well, such as the peripheral neuropathy mentioned in that review.