My great-grandfather, James C. Siegel of Schenectady, New York (30 AUG 1884 - 13 DEC 1951) was married to a Catharine Pearl Chadwick on June 30; I don't have the year and this information from a newspaper announcement my dad had. When James C.Siegel died, he was married to Frances (18 FEB 1891 - ?). James Siegel had a son named William Matthew Siegel (1 APR 1907 - 26 APR 1970). It appears Frances Chadwick was William's mother; according to William Matthew Siegel's death certificate. Is it possible she was once Catharine Pearl and then changed her name to Frances? I'm not finding very much information for either name.
The best fit I can find to the people in your question are entries from the 1910, 1920 and 1930 US censuses on FamilySearch. All are in Schenectady, New York. Assuming this is the right family, we can list the most relevant-looking fields:
1910 James Siegel Age 25 Married 3y Born: NY Father born: Germany Catherine Siegel Age 19 Married 3y Born: NY Father born: NY William Siegel Age 3 Born: NY Father born: NY 1920 James Siegel Age 35 Born: NY Father born: Germany Frances Siegel Age 29 Born: NY Father born: Ireland 1930 James Siegel Age 45 Marr @ 22yo Born: NY Father born: Germany Frances Siegel Age 36 Marr @ 18yo Born: NY Father born: NY William Siegel Age 23 Born: NY Father born: NY
The records for James Siegel match well. The 1910 and 1930 censuses say he (first) married at the age of 23, and William must be his son with Catherine. A birth certificate (or baptism record) for William would clarify that, of course.
James is married to Catherine in 1910 and Frances in 1920 and 1930. Could they be the same person? That's more difficult. Frances' stated age rose by only seven years between 1920-1930, so at least one of those ages must be in error. Her father was born in Ireland, or in New York. Again, one of those can't be right.
The 1910 census tells us that James and Catherine married 3 years earlier, so around 1907, when James was 22 and Catherine was 16. The 1930 census says that James married at age 22, around 1907. That's a good match. It says that Frances married at age 18. If we take her age at the stated 36, then she married in 1912, too late to be William's mother. If she was actually 39, to match the 1920 census age (and the 1891 birth date in the question), then she married in 1908. That puts her closer to Catherine in marriage age and date, but there are still errors of 1-2 years. Some cause for caution, but for census entries those aren't particularly awful discrepancies.
Given the several apparent errors, it's hard to know what data to trust to reach any reasonable conclusion. Frances is a fair, but not perfect, match to Catherine, so if we default to scepticism we would start by treating them as different people.
In that case, look into the possibility that James remarried, to Frances, within a few years of William's birth. The likely reason for that would be that Catherine died some time between 1910 and 1912. Look for that record, and for James marrying Frances soon after. If the records don't exist, then that may lend more weight to Catherine and Frances being the same person with a simple name change.
In the question you give both Catherine and Frances the (pre-marriage?) surname of Chadwick - where do those details come from? Any extra records you have are worth stating by editing the question, as they could be very relevant. If they did have the same surname, that must make it more likely that they were the same person, but it's not guaranteed. For example, my tree has a man who married his late wife's niece, who had the same surname as her.
I'd note also that a William Siegel, born 1907, has 1915 and 1925 New York State census records (for Schenectady) returned in the FamilySearch search, in which he is described as "Grandson". I don't have paid access to the Ancestry.com site to see the full results, but if you do then those must be worth a look to see if it's the right William. If you can find the grandparents, and they are not called "Siegel", then you may have found Catherine and/or Frances' parents, which should help find other records.
Its not very likely they were the same person. Changing a first name happens once in a blue moon but changing a second name too doesn't really happen. It is more likely that your grandfather married twice.
Best bet (I don't know what site you are using) is to try and find two seperate marriage certificates, one for him to each of the women. If you can't find that, then you can begin exploring the avenue of a possible name change. I have never tried, but I'm assuming there are record offices out there for keeping track of such things.