When a board says no, an acquirer has two routes: a tender-offer to buy shares directly, or a proxy-fight to replace the board with directors who will approve a deal. Both take months and both are public, so the target's price stays elevated on takeover speculation.
Defences are well developed. A poison-pill, a staggered board, and a white-knight search can all be deployed, which is why most hostile approaches eventually become negotiated deals at a higher price or fail outright.
Example: a bidder offers $44 against a $34 market price and the board rejects it. The stock settles at $39, pricing roughly a 50% chance of a deal at $44 and a 50% chance of a return to $34.
Related: poison-pill, proxy-fight, white-knight, tender-offer