You're really probably better off thinking about what audible alarms nurses use - in addition to the dials - for human patients. A more analogous situation.
The plethora of alarms in aircraft are due to (at least) four long-term developments:
- Increase in the number and complexity of the systems to be monitored
- Reduction in the number of pairs of eyes available to monitor them (from 4 to 3 to 2 over 50 years)
- speed at which "situations" can become critical at 500 kts.
- air crash experience, which led to questions of "Why didn't the crew notice X, and what can we do to improve their awareness (or at least look as though we improved it)?"
Crews have been "overloaded" by the alarms in some situations where there are multiple related failures. Many/most times they are able to sort things out - but on occasion they've gone into "deer in the headlights" mode.
....or not even heard them, or filtered them out, as Ka8 mentions.
See the crisis moment in the film Apollo 13 for a fair representation of what "alarm overload" can look/feel like.