The risks associated with advanced maternal age are very much real. I'd put it more at 39 than 35 that the risk jumps the most, though.
Advanced maternal age is strongly associated with chromosomal abnormality (and incidentally advanced paternal age shows some association with single-gene disorders), and the reason for this is the way that egg production differs from sperm production. I'm sure most of you will remember being taught at school that meiosis results in 4 haploid gametes, but that actually is only true for the production of sperm. Sperm are produced constantly starting from puberty and meiosis in males does result in 4 haploid gametes. Eggs are different.
Eggs are produced in utero. They enter the first phase of meiosis in utero. At the stage where homologous chromosomes pair and cross over, the points of cross over are held together by chiasmata, which keeps the pairs together. The role of crossing over/chiasmata formation here is actually to hold the chromosomes together, not for genetic diversity (though it contributes, it's not the purpose -- independent assortment is the major contributor to genetic diversity). In females, meiosis arrests here in utero.
This means that a woman is born with all of her eggs already half way through meiosis, and the chiasmata are responsible for holding those chromosomes together until the egg is released during the menstrual cycle, at which point the egg will exit meiosis I and begin meiosis II, forming a single gamete plus the first polar body. Meiosis II and the second polar body only occur after the egg is fertilised. So meiosis in females only produces one gamete and only completes on fertilization.
The drawn out nature of the process makes it error prone. Chiasmata weaken over time. Loss of chiasmata integrity significantly increases with age and results in chromosomal abnormalities because where there are chromosomes that are not held together any more, when meiosis I completes, you will have aneuploidy or trisomy due to nondysjunction of homologous chromosomes. Additionally, errors in meiosis II also lead to trisomy due to nondysjunction of the sister chromatids.
Trisomy 21 or Down Syndrome appears to be associated with advanced maternal age more than other errors from nondysjunction, but that's not actually the case. The reason it's seen as more common is that it's one of the few viable outcomes of nondysjunction. Most will be incompatible with life and will be lost either as miscarriage that is obvious or before the woman even knows she was pregnant. This is why it's harder to conceive as you get older. The percentage of eggs which will have errors in meiosis gets larger as we age.
So the risk is real. It is also probably not overestimated since a large number of failed pregnancies won't be noticed or reported, and where early miscarriage occurs the reason won't necessarily be determined.