It depends. There is no golden X:Y ratio but in the end it boils down to this: a department is staffed correctly when its comprised of reasonably competent people that can get their work done in a standard work week without creating an ever-growing backlog.
To expand on that, the workload for a department should be reasonable, which means a 40-hour work week with zero overtime should be the rule for most of the year. Requiring people to put in reasonable amounts of overtime is fine but it should never be systemic. Additional hours are much less efficient and sustained overtime will cause burnouts, cost significant amounts of sick leave and, unless that time is compensated in some way, will push out high-performers who have plenty of opportunities elsewhere.
You need reasonably competent people staffing it because low performers will get much less done and can end up increasing everyone else's workload. Managers should be building a team that's more than just competent, but that takes time and resources. High performs will get more done but also shouldn't be worked to exhaustion.
A backlog is created when tasks that should be getting done aren't. It's normal for workload to shift over time and for low-priority tasks to be delayed during crunch time, but a department should be able to work through their backlog during slower times. If the backlog keeps growing and there is no slow period that's a sure sign of an understaffed department, whether that's in the quantity or quality of employees.
You're asking about an HR department but that question is even more difficult because the tasks assigned to HR vary hugely across companies and countries. As an example, Ricardo Buettner suggested the following as HR's core responsibilities:
- job design and analysis,
- workforce planning,
- recruitment and selection,
- training and development,
- performance management,
- compensation [payroll]
- legal issues
In large companies, each of these can be a full-time job. Start-ups, even with dozens of employees, sometimes don't have even one HR person on staff and payroll duties are outsourced or assigned to management. Arguments can be made in any direction when it comes to whether a given task should fall to HR or to the separate departments/managers of a company. Generally though, it's both financially and organisationally rewarding to split off as many personnel management tasks as possible to HR.
Looking at your example, 2 HR for 150 staff can be perfectly reasonable if all they did in the company is process an automated payroll and handle the occasional issue or complaint. If they had low turnover they'll even have time to organise interviews or do the new-hire paperwork. It's when you start expanding your interview process, delegate more and more work to HR, start demanding more accuracy or reporting or end up with poor performers in the team that they'll become understaffed. Note that it's difficult for an outsider to say what precisely went on. Maybe their responsibilities shifted or increased. Maybe the previous team were unsung heroes and put in 60 hours a week. Maybe the new guy is incompetent.
Your head-scratching response is about as much as you can say. If the HR director's manager has doubts about the personnel requirements then he should take those up with him directly and ask for justification for the new hires.