Governor Ricketts did not respond to a request for comment on Monday, but has for many years maintained a position similar to that of Mr. Peterson. In a column posted online before the 2016 statehouse debate, he called marijuana “a dangerous drug” and said any form of legalization could be a slippery slope.

“We must be cautious before we follow the lead of other states,” he wrote. “Legalization of marijuana for any purpose has proven to be a risky proposition because the controls placed on its use in other states have fallen short.”

Matthew Schweich, the deputy director of the Marijuana Policy Project, which has backed the Nebraska effort, said advocates there could learn from successful medical marijuana efforts in conservative states like Utah.

“You’ve got to explain to voters that medical marijuana is a question of compassion and of allowing sick and suffering people to treat their conditions without being treated as criminals,” said Mr. Schweich, who added that his organization was considering 2020 medical marijuana campaigns in Idaho and Mississippi.

“You have to emphasize that medical marijuana is not the same thing as full legalization of marijuana” which has been passed in only 10 states, he added. “That was the top argument used against us in Utah, that it was a form of full legalization or it would lead to full legalization, and I expect that to be the main argument used by the opposition in Nebraska.”

Ms. Wishart, who sponsored the legislation in 2016, said she would introduce another medical marijuana bill next year “to give the Legislature one more chance,” but she said she was “not that optimistic.” But she pointed to a recent history of progressive policy victories at the ballot box, including Medicaid expansion in November and a minimum-wage increase in 2014, as a reason to be hopeful.