You can't have perfect accuracy. Your response variable seems not a rare event neither.

Can you report the frequencies in the crosstable of predicted vs observed responses?

What is AUC and an Rsquared of your models?

This is not a problem of "too few unsuccessful events" for a logistic regression model.  An "event" in 16% of the dataset is plenty, especially in a dataset this large (>30,000 observations).  The problem could be that your independent variables are just not very strongly associated with your dependent variable.  Have you looked at some simpler descriptive statistics before proceeding with a logistic regression?  For example, look at the percentage of "unsuccessful" events across different categories of independent variables?  It is possible that your independent variables just do not "predict" very well.

Different categories of independent variables?  I dint really get it. My independent variables are numeric.

Or you are talking about extract the rows according where my dependent variable is equal to 1 and study that particular pattern in the data?

"Different categories of independent variables? I dint really get it. My independent variables are numeric."

Then you should group them into a few basic bins (quartiles would be fine) and get some idea of how the percentage of "failures" or "unsuccessful events" compares across the quartiles.  Just something to give you an idea.

For example, suppose that I have a simple Yes/No outcome (death in hospital) and I want to know how that varies by age.  To get a basic idea for how well age "predicts" my outcome, I can bin age into quartiles.

Suppose that the failure rate is 6% in Q1, 7% in Q2, 8% in Q3, and 9% in Q4.  If my dataset is very large, it is possible that I will see a "statistically significant" relationship between age and mortality, but that variable alone will not "predict" mortality very well; even some people in the lowest age quartile died, and many people in the highest age quartile lived.

Anyways, that's what I mean.  Get a sense for how much the rate of "failure" varies across the range of your independent variables.  

Dear Andrew, thank you so much for the response.

"Get a sense for how much the rate of "failure" varies across the range of your independent variables."

I have got couple of doubts:

1) Should I run this analysis with categories of one independent variable or combination of independent variables? (I am sure that it must be one independent variable at a time , but just want to confirm) 

2) Suppose I observe that my failure rate  is negligible to none for first two quartiles and very significant in higher two quartiles . In this case while I am building the model should I discard the data in lower (first) two quartiles ? 

"1) Should I run this analysis with categories of one independent variable or combination of independent variables? (I am sure that it must be one independent variable at a time , but just want to confirm)"

Yes, the purely "descriptive" analysis that I have suggested, you should begin with one IV at a time.  This is not a final solution - merely something you should do to better understand your data.  I am so sick of people who just start creating logistic regression models that have glossed over the absolute most basic descriptive analysis to determine how their IV"s are associated with their outcome variable BEFORE starting the modeling process.

"2) Suppose I observe that my failure rate  is negligible to none for first two quartiles and very significant in higher two quartiles . In this case while I am building the model should I discard the data in lower (first) two quartiles ?"

NO!  You should not discard the data - this is going to be part of building a good model for prediction.  If there is one factor which can be used to "rule out" having a failure, that alone will be useful in giving your model some predictive power.

Andrew D Althouse I have a sample of fifty thousand individuals and out of them only 1200 participate in markets making the response rate of around 3%. Would this count as a rare event in logistic regression?

Given the context of market participation, it would not be theoretically correct to call this as a rare event though statistically it can be said so.

Further, shall I use a sub-sample of only those eight thousand individuals who have income above certain limit and analyze 1200 participants among these 8000. Here response rate is around 15% so it won't be rare event but the predictive power of the model suffers. Can you provide some suggestions?

Thank you!

Can try skewed logistic regression in stata.

https://www.stata.com/manuals13/rscobit.pdf

Thank you for sharing, sir.

It seems to solve the purpose. Thank you.