Can Down syndrome occur without trisomy 21 in the karyotype of an individual? Or vice versa can a person have trisomy 21 while unaffected by Down syndrome?
First question: No, all Down Syndrome patients have a inappropriate duplication of part or all of chromosome 21.
Second Question: More complicated. The phenotype can vary in severity depending how much much of chromosome is duplicated, whether it is duplicated and stuck onto the end of chromosome 14 (translocation vs. non-disjunction). Lastly, a rare form of this event is defined by incomplete penetrance or mosaicism of cells that contain duplication. Short Answer: No, In rare cases, a carrier can pass on trisomy 21 without being affected themselves but they themselves are a "balanced carrier" and don't have extra chromosome 21 material. See last paragraph below from the NIH and this quote from the Mayo Clinic:
When translocations are inherited, the mother or father has some rearranged genetic material, but no extra genetic material — this means he or she is a balanced carrier. A balanced carrier has no signs or symptoms of Down syndrome, but he or she can pass the translocation on to children, causing extra genetic material from chromosome 21
Types of Down Syndrome details from: NIH Down Syndrome Information Page
Research shows that three types of chromosomal changes can lead to Down syndrome.
Complete trisomy 21. In this case, an error during the formation of the egg or the sperm results in either one having an extra chromosome. So after the egg and sperm unite, the resulting cells will also have three copies of chromosome 21. The complete extra copy of chromosome 21 is in all of the person's cells—or a complete trisomy. Complete trisomy 21 is the cause of about 95% of Down syndrome cases.1,2,3
Mosaic trisomy 21. Not every cell in the body is exactly the same. In about 1% of Down syndrome cases, most of the cells in the body have the extra chromosome, but some of them don't. This is called "mosaicism." Mosaic trisomy 21 can occur when the error in cell division takes place early in development but after a normal egg and sperm unite. It can also occur early in development when some cells lose an extra chromosome 21 that was present at conception. The symptoms of someone with mosaic trisomy 21 may vary from those of someone with complete trisomy 21 or translocation trisomy 21, depending on how many cells have the extra chromosome.1,2
Translocation trisomy 21. In this type of chromosomal change, only part of an extra copy of chromosome 21 is in the cells. The extra part of the chromosome gets "stuck" to another chromosome and gets transmitted into other cells as the cells divide. This type of change causes about 4% of Down syndrome cases. There are no distinct cognitive or medical differences between people with translocation trisomy 21 and those with complete trisomy 21.
Sometimes, a parent who does not have Down syndrome may carry a translocation in chromosome 21 that can be passed on to children and cause Down syndrome. Studying the parents’ chromosomes can reveal whether this is the cause of the syndrome. A genetic counsellor can assist families affected by translocation trisomy 21 in understanding the risk of Down syndrome in future pregnancies.1,2,3
Also important to note that people can have trisomy 21 develop in somatic cells spontaneously, later in life without having most of the effects of DS. But this can significantly predispose to leukemias shown here, which is an interesting area of study. Thanks to @The Nightman.