All of the items on my application for a postdoc position was completed around 2 weeks ago. The prospective PI told me he will take a look over everything and gets back to me in a day or two, he did not get back to me and I sent a follow up email, he replied very quickly with a nice message and said he could not take a look and I should remind him again on the weekend. I reminded him over the weekend and he got back to the Monday afterwards, he again sent a nice message and apologized and said he still could not take a look but he will do it today or tomorrow, but it did not happen. I again send a follow up email on Thursday and he replied to me quickly and said he is just left behind and he needs to take a look over everything just one more time and that he will do it the same day, however he did not get back to me. How should I navigate this situation? Is this good sign or a bad sign? Any tips on how to follow up? Thank you!
My guess is that he is just incredibly busy. I'd suggest you be patient for a bit. Don't give up on other opportunities and he should respond in time.
However, also take a look at what this might mean for the long run. Being very busy can be good or bad. Sometimes it implies productivity which can benefit you.
But unless you have a reason to need to know quickly, let it go for a bit.
In certain points in the academic calendar, my own respinsibilities and priorities are this heavy. I have every intention of reviewing an application, but other things get in the way. Sometimes, I don't even have time to write an email, eat a meal or even sleep!
Trying to read signs in situations like this is no better than reading tea leaves -- you will see what you see. Until you receive a final answer, I suggest taking the PI's message at face value and send a follow-up email. It doesn't have to be long or circuitous:
Hello, Dr X. I contacted you yesterday about my application. You responded by asking me to check in today. I was wondering whether there was any progress.
That the PI didn't respond to your last email might be due to several factors, not all of which are related to your application. In that case, I would write something like
Hello, Dr X. I didn't receive a reply from you and wanted to check that you received my email about reviewing my application for position yyyyy. I'm still interested in the position and will wait patiently for your reply.
and I should remind him again on the weekend.
Huge red flag, potential PI is a person that think he will do whatever unrealistic goal of the week by Friday noon, then understands they will not manage, so why not extend the working days to the weekend (just for this one time, of course ... which translates to "hey, sometimes I work on the weekend, but I care for my personal wellbeing and I do so only once per week!").
Potential PI is putting his/her own free-time below your anxiety for getting considered for a Postdoc position ... 3 months working with they and they would unconsciounsly expect the same from you.
Do not follow up, let the process go on its own, my reccomendation is to consider any submitted application to job/academic position/research grants as rejected unless proven contrary ... I know that statistically I am right, plus it will leave you constantly thinking for a realistic plan B/C/D/.../Z which is the only sane thing to do in an high-risk (in terms of opportunities and satisfaction) working environment as the academia.