MrsPhD

This thread is for FSU. You should not ask questions in the wrong thread.

That is more an Econ thing because they admit big cohorts and then kick out a lot of people. I have known of PhD programs in Political Science in which 40-50% of a cohort was kicked out for failing to pass comps, but it was not a regular thing, just something that happened once. Most departments like Duke or NYU have small and the people that leave are those that (1) realize that they don't want to do a PhD (2) don't study or put any effort and are asked to leave (3) fail comps (if you are already working with someone and make an impression, is less likely that you will fail comps, unless you really do a bad job and it is not a "bordeline" case; usually, you can retake comps once). I've found (3) to be the lower case of attrition. 
 

It is hard to estimate attrition because it is not the case that everyone on the market is from the same cohort. Students go at different times because some take longer than others. Also, not everyone goes on the job market because some decide not to go into academia, others get postdocs before getting listed on websites, etc. 
You can ask about attrition in visits. I know people that went to Duke and NYU, and there is nothing out of the ordinary with attrition. They are not kicking out people for no reason.

You are being naive. 
Both programs are similarly ranked and with similar placements. At this point, what will matter is the effort the candidate makes, not a general difference between Duke/NYU. Also, it is hard to know placement in 5-6 years. 
What could matter more is placement by subfield and advisor(s). Maybe this particular candidate has an interest in something that program X is better than program Y. 
 
 

I'm not sure if Duke is more prestigious that NYU. It is similar.
Duke has a good methods sequence/training as well. There are a lot of people who do advanced methods that went to Duke and some methodologists as well. As far as I know, the methods training/focus is different. At Duke, students take classes in the stats/computer science department, so they focus on statistics, Bayesian statistics, machine learning, etc (Duke has one of the best stats departments in the country). NYU focuses more on causal inference and experiments, though you have Spirling doing text analysis and Rozenas is a Duke PhD so he does more statistical modeling. 
Your focus should be more on your substantive interests (American, Comparative, IR) and look for a couple of people who could be your advisors (so associate level, maybe full professor as long as they are not retiring, and some people that are about to get tenure). Try to ask them about their current projects and see if they co-author with graduate students. You can also try to talk to their students during the visiting weekend. Ask about people leaving/retiring (I know J Larson is leaving NYU, for instance).
 
 
 
 

I'm not sure if Duke is more prestigious that NYU. It is similar.
Duke has a good methods sequence/training as well. There are a lot of people who do advanced methods that went to Duke and some methodologists as well. As far as I know, the methods training/focus is different. At Duke, students take classes in the stats/computer science department, so they focus on statistics, Bayesian statistics, machine learning, etc (Duke has one of the best stats departments in the country). NYU focuses more on causal inference and experiments, though you have Spirling doing text analysis and Rozenas is a Duke PhD so he does more statistical modeling. 
Your focus should be more on your substantive interests (American, Comparative, IR) and look for a couple of people who could be your advisors (so associate level, maybe full professor as long as they are not retiring, and some people that are about to get tenure). Try to ask them about their current projects and see if they co-author with graduate students. You can also try to talk to their students during the visiting weekend. Ask about people leaving/retiring (I know J Larson is leaving NYU, for instance).
 
 
 
 

Some with money do give more benefits to people they particularly want to recruit. I've heard of people that have negotiated at UCSD. But I'd be careful/very polite because some schools -- public and private -- make the same offer to everyone in order to promote collegiality and fairness.
That said, some public schools have special fellowships, but those are usually part of a special contest awarded at the university level and nominated by the department.
Some departments do have the possibility of putting you in RA duties rather than TA duties, or they might be able to give you like a small research fund to get reimbursement for books, travel to conferences, or field work. If there are centers affiliated to the political science department, those might be able to give you a bit more money, though that could come with strings attached. You could inquire about summer funding (most schools provide 9-month stipend, but you still have expenses during the summer). 
 
 
 
 

Some with money do give more benefits to people they particularly want to recruit. I've heard of people that have negotiated at UCSD. But I'd be careful/very polite because some schools -- public and private -- make the same offer to everyone in order to promote collegiality and fairness.
That said, some public schools have special fellowships, but those are usually part of a special contest awarded at the university level and nominated by the department.
Some departments do have the possibility of putting you in RA duties rather than TA duties, or they might be able to give you like a small research fund to get reimbursement for books, travel to conferences, or field work. If there are centers affiliated to the political science department, those might be able to give you a bit more money, though that could come with strings attached. You could inquire about summer funding (most schools provide 9-month stipend, but you still have expenses during the summer). 
 
 
 
 

I think this is a somewhat wrong approach.
The goal should always be what do I need to do to get into the best program possible. Not, conversely, I didn't get into X programs this cycle, how can I change which schools I apply to so that I receive acceptances?
Yes, we all want acceptances; but we should strive for acceptances from programs that are going to give us the best chance of getting a job.
What I would do in your shoes:
- First all, retake the GRE. Then retake it again, and again, until you have at least high 150s or 160+ in your quant section. You're making it easy for schools to reject you if it's not that high. Everyone is capable of getting at least 75th percentile in the quant section, just takes hardwork and discipline. 
- Critically assess every part of your application, and fix whatever is weak. Rewrite your SOP, get some research experience, get a working paper in order from your master's thesis that you can submit as a writing sample, continue to build relationships with your letter writers, ect. 
I usually trust applicant's assessments of which programs they think are a good fit for them research wise (although, sometimes, people really miss the mark here). I doubt the problem is which programs you applied to, it's that you lost out on the competition battle - which happens for everyone, by the way.
Lastly, don't pigeon hole yourself. There's nothing inherently wrong with being more qualitatively inclined...but there are certain questions that cannot be adequately answered without quant methods. A lot of IR students/scholars are using formal models to build their theories. You may find out that you can measure your variable using text analysis of documents. Don't shy away from this...and do not apply to programs specialized in qualitative methods; it makes getting a job even more of an uphill battle. 
It also may be helpful if you provided a list of the programs you applied to this cycle.