If Miguel went to science club every Thursday for 4 months and three weeks and each Thursday 25 chairs were set up for club members to sit in, how many times would Miguel have sat in a seat that Clara had already sat in before?
If Miguel went to science club every Thursday for 4 months and three weeks, this would be a total of (4 months) * (4 weeks/month) + 3 weeks = <<4*4+3=19>>19 weeks.
If there were 25 chairs set up each Thursday, the total number of seats available during this period would be 25 chairs/Thursday * 19 weeks = <<25*19=475>>475 seats.
Since Miguel attended every Thursday, he would have sat in a seat that Clara had already sat in before 475/2 = <<475/2=237.5>>237.5 times.
However, since we cannot have a fraction of a seating, the answer is rounded down to 237 times. Answer: \boxed{237}.