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