You could use another variable to keep track of the answers given by the user:
#This is the size of the array
YEAR_SIZE = 12
months = [] ###This is the array that will hold the rainfall for each month
monthNames=['January','February','March','April','May',
'June','July','August','September',
'October','November','December']
def getMonthlyRainfall():
ANSWERS = 0
while True:
try:
total = 0
for month in range(ANSWERS, YEAR_SIZE): ###This loop iterates 12 times for 12 entries
print ("Enter the rainfall for",monthNames[month], "in inches")
x = input()
months.append(float(x))
ANSWERS = ANSWERS + 1
except:
print ("Try again")
getMonthlyRainfall()
In this case ANSWERS