If the false positives would have applied to you, then avoiding them is OK - but you don't know this in advance. That is why there is a sequence of tests, first a mammogram which is (relatively) inexpensive and highlights possibilities, then those individuals highlighted get further tests to be certain. Some might have been spared further tests had they refused the mammogram, but others might not be diagnosed at all.
What it boils down to is, would you rather be called back for more tests and then find you DON'T have breast cancer, or ignore the mammogram appointment and then find that you DO have it?
"Mammography is good at finding breast cancer, especially in women ages 50 and older.
Overall, the sensitivity of mammography is about 84 percent. This means mammography correctly identifies about 84 percent of women who truly have breast cancer.
Sensitivity is higher in women over 50 than in younger women. It is also higher in women with fatty breasts compared to women with dense breasts.
The chance of a false positive result is higher among younger women and women with dense breasts. (Most women younger than 50 have dense breasts so there is overlap among these groups.)
Getting a false positive result can cause short-term fear and worry. However, these feelings do not seem to have lasting effects.
Remember, most women called back for a false positive result do not have breast cancer."
ww5.komen.org/BreastCancer/AccuracyofMammograms.html