Table partition should not be selected
I cannot fully answer this question but I can give you some tips on what to look around for.

First thing I would do is check the dates inside each partition as follows

SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p13q1) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p13q2) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p13q3) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p13q4) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p14q1) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p14q2) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p14q3) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p14q4) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p15q1) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p15q2) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p15q3) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p15q4) GROUP BY date_x;
SELECT date_x,COUNT(1) datecount FROM TABLE_X partition (p16q1) GROUP BY date_x;


If dates are partitioned correctly, then you have a bug-infested mysql in relation to the query optimizer. If dates appear in a partition that do not belong, I would reload the table from scratch as follows:

CREATE TABLE TABLE_Y LIKE TABLE_X;
INSERT INTO TABLE_Y SELECT * FROM TABLE_X;
ALTER TABLE TABLE_X RENAME TABLE_XXXX;
ALTER TABLE TABLE_Y RENAME TABLE_X;


Then, retry your queries.

I noticed you are using MySQL 5.1. There are many patches to the partition code (such as ALTER TABLE ... REPAIR PARTITION). I would suggest installing MySQL 5.6 on a test server, copying the data into it and try out your queries on.