Hedging removes a risk you did not want; it does not remove risk. Delta hedging an option leaves gamma, vega and theta - which is the position you actually intended.
Basis risk is the standard failure: hedging with a correlated but non-identical instrument leaves residual exposure, and correlations tend to break precisely when the hedge is needed.
The cost. A perfect hedge eliminates the profit along with the risk. Every real hedge is a decision about which risks to keep, not an elimination of risk.