What is the biggest sign that a new hire is not going to work out on your team?
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I think that many signs, as higlighted by Murray, could turn some lights yellow or even red.
There are 3 more I would add:
1. The overconfidence: "I know how to manage this, I've done it at my previous job before and I was the best."  This is always wrong unless you're facing someone with deep experiences.
2. The skeptical on the processes: "I think the way you're doing this isn't adequate.  I've always been doing this another way."   This is bad for the team chemistry if you had common and standardized ways of doing things.
3. The misplaced position: "I was a manager before.  I hope a new manager position will open soon, meanwhile I'm ok doing some architecture with you guys."   This means the individual is constantly looking to switch job and he will not be dedicated to the tasks you had given.  Plus, it might encourage others to look for other positions!
Those might be deeper issues than others and it's very hard to correct them.  But it worth the effort to try for sure.
In my experience they are having severe issues when they go away and do a task and they
1. Do not volunteer any updates.
2. Does not ask enough questions.
3. Every status update is all good news.
I think these are all correctable, but you have to catch them early before they lose confidence in themselves and the process.

My response to the question, which is a good question, is that do not be too quick to judge. Depending on the organization and the role, the learning curve can be longer. If the employee has been in the role for a considerable amount of time, then I believe that the signs that a new hire will not work out long-term are:
1) Resistant to change, more negative with new policy and process rollouts that undermine change adoption, and an unwillingness to embrace new business practices.
2) Not a team player, withdrawn from the team, demonstrates a lack of interest in nurturing healthy strategic partnerships.
3) Unwilling and unable to perform the duties and responsibilities of the role, and poor performance/results. Does not respond to being held accountable.
Lastly, you should also examine if the individual is an excellent cultural fit. Even if performance is to expectations, poor cultural fit can present longer term issues.