No. Many companies have hiring rate of 10% [1] which means hiring 1 candidate may cost you 10 person-days assuming full day interviews for each candidate [2]. You can probably double that to cover the cost of phone screens, so let's say 20-person day of work per good hire. Assume you had 4 false negatives just because you are so aggressive and so a good hire ended up costing you 100 person days instead. This cost is still
tiny compared to damages that a true negative would cause otherwise, i.e.,
(1) loss of entire person year or even two because annual reviews need to accumulate evidence for HR to fire (may be less time if you were startup without real "HR")
(2) amount of cleanup other people have to do after that new bad hire
(3) loss in moral for good people in your team who now perceives your hiring process at the company as "broken"
(4) delays + bugs introduced in product because actual work probably didn't got done or badly done despite of you filling up your headcount
(5) amount of money lost in salaries, signing bonuses, office space and benefits (typically > $200K)
(6) amount of productivity lost because of wasted time by good people in the team trying to "ramp up" your true negative
(7) emotional stress you caused to good people wondering them about their job stability and to managers who wasted their time in months of paper work and lot of explaining
(8) emotional stress you caused to your true negative being fired who had moved across the country for you, bought a house on mortgage and had 3 school going children
(9) Most likely, if you are big company, true negative didn't actually got fired because hiring manager never wanted to admit it. S/he was encouraged to join another team or role or even learned political tricks to get promoted contributing to ongoing bozo explosion[3]
(10) I could go on and easily justify probably 3X-10X loss compared the case if true negative was avoided
Footnotes:
1 - Many companies specify their hiring rate over total resume they received which is wrong. I'll use 10% as total number of full interviews that needed to be conducted which is average of 5%-15% at most companies.
2 - This is bad math. Assuming random trials, it would be actually 5 person-day on average but intuitive approach doesn't produce entirely bad results here so we will go with that.
3 - http://guykawasaki.com/how_to_prevent_/