It's extremely unlikely.
The idea that a man could produce "only one type of sperm" is absolutely false, it's totally impossible and anyone thinking this needs to go back and redo year 9 biology.
The vast majority of men have XY chromosomes. When sperm is created the chromosomes are split in half. There will always be equal numbers of X sperm and Y sperm created. It's not possible for a man to create no Y sperm - he would need to be XX chromosome, in which case he'd be female. It's not possible for him to create no X sperm - he'd be YY chromosome, which doesn't exist, a zygote formed with this combination would be rejected as too much genetic material would be missing.
There are some genetic mutations which result in a man being XXY or XYY - these are rare, and their sperm will still be produced with both X and Y chromosomes (I believe in this case, the "extra" chromosome is passed on in the sperm and may be inherited, but these conditions are usually caused by a random mutation). XXY is also associated with fertility problems. XYY is less so as I understand, but it wouldn't produce twice the amount of sperm with Y chromosome. It would produce a normal amount of X chromosome and then either all the Y chromosomes are also YY or a proportion of the Y chromosomes are YY and some are Y. (If I understood this correctly!)
The specific chromosomes that are present in a sperm cell do not affect its ability to reach the egg, that isn't possible. However, it is possible for a man to carry a genetic defect on either the X or Y chromosome which causes fertilised eggs to be rejected by the body or causes miscarriage. So it is possible in rare cases for a man to produce only VIABLE sperm of one sex. But he will still produce sperm with the opposite chromosome in equal numbers, it's just that they won't result in viable embryos.
And, again, this is rare.
OTOH, the chances of producing all one sex just by chance are very high, and if you already have several the same sex then it's always a 49/51 chance - which is extremely high. Therefore people come to incorrect conclusions because they have a "hunch" (oh it must be he can only make boys/girls) and then it's proven right on essentially a coin flip, so they cement this idea when actually the highest likelihood is it's just chance.
Think about it - people rarely claim this before their first baby and then it's totally right. They usually claim it when they already have two (or more) of the same sex, and then they have a 50/50 chance to get another of the same sex.
Chances of having 2 children the same sex: 50% (2/4 possible combinations = 1 in 2)
Chances of 3 children the same sex: 25% (2/8 possible combinations = 1 in 4)
Chance of 4 children the same sex: 12.5 % (2/16 poss combo = 1 in 8)
Etc. The chances of having one of the chromosome disorders are roughly 1 in 500. It's only after having 10 children the same sex in a row that the chromosome disorder is more likely than random chance.
For typical numbers of children in a family (2 up to about 4 or even 5) the likelihood of it being random chance is much much more likely than genetic disorder.