"I have been threatened by text messages from my estranged family today for having spoken out-- for speaking the truth," the 57-year-old Saffo told TooFab on Friday.
Saffo said her older brother, Earl Brosnahan, texted her, "A public example could equally be made of your private missteps. Be very mindful of what you say."
Brosnahan, whose daughter is Marvelous Mrs. Maisel star Rachel Brosnahan, issued a statement in response, telling TooFab it is "sad and very hurtful" to see people with "no real knowledge of the situation" giving "false, speculative information that maligns Kate's character and belittles the health issues she bravely fought."
"We once again ask that the family's privacy be respected during this challenging time," Brosnahan's statement concluded.
"She was always a very excitable little girl and I felt all the stress/pressure of her brand (KS) may have flipped the switch where she eventually became full-on manic depressive," Saffo wrote of her sister. "I'd come so VERY close to getting her to go in for treatment (to the same place Catherine Zeta-Jones went for her successful bipolar treatment program). I'd spoken with them on the phone (not telling them exactly who the patient would be). They agreed to fly in and talk with her and take her with them to the treatment center."
Saffo claimed Spade "was all set to go" but then "chickened out." She said Spade's husband, Andy Spade, helped in trying to get Spade treatment and made plans for taking care of Spade's daughter, Frances Beatrix, but "nothing ever came of it."
"She was a dear little person. So dear — so kind, so funny. I'll miss our 6-7-hr-long phone conversations between NY and NM," Saffo wrote. "I'm off to bed for a good cry."
However, a Spade family source told PEOPLE that Saffo is estranged from the family. They are "disgusted and saddened that at this time of great sorrow, Kate's sister who has been estranged from the entire family for more than 10 years would choose to surface with unsubstantiated comments."
"Her statements paint a picture of someone who didn't know her at all," the source said.
In addition, Saffo's claim that Spade did not want to be treated contracted Andy's Wednesday statement.
"She was actively seeking help and working closely with her doctors to treat her disease, one that takes far too many lives," Andy said of his wife.
Spade was found dead in her Manhattan apartment on Tuesday at the age of 55.
If you or someone you know needs help, please call the National Suicide Prevention Lifeline at 1-800-273-TALK (8255).