(FORTUNE Magazine) – When President Clinton in late November asked AMR's American Airlines to sit down and work out a settlement with its striking flight attendants, their union's President Denise Hedges hailed ''a great victory,'' and pundits predicted a union comeback. For once, the experts may have a point -- in the short run, that is. Emboldened by how quickly American CEO Robert Crandall agreed to arbitration -- and despite the passage of NAFTA, which labor mostly opposed -- unions have a new sense of resolve. Other companies facing labor negotiations -- among them UAL's United Airlines and Caterpillar -- may well have to ride out some long, bitter strikes to prove they have the will to keep labor costs in line. Unions still face some big problems, particularly diminishing worker support. David Macpherson, an economist at Florida State University, says that union membership in the private sector shrank by 199,000 last year, to 9.7 million. That's about 11% of the U.S. work force and the lowest level since the 1930s. Adds Daniel Mitchell, a professor at UCLA's business school: ''With the number of workers covered by unions declining and no new money going into organizing, management will still hold the upper hand.''