This depends on what you mean by inevitable. Most people in the Soviet Union would not have thought, or been told, that the USSR was going to come apart, because the leaders of the USSR (primarily referring to Gorbachev) did not think the USSR would come apart. With that said, there was nothing the leaders were in this case keeping from the people.

However, many people may have cast their doubts on the Soviet Union in its waning years, due to increasing economy hardship, and a rapidly declining economy. One of Gorbachev's main policies in his time as leader of the USSR, was Glasnost (meaning "openness"). This involved the government divulging its information with the people, and making politics more transparent in the USSR. Perhaps most importantly, there was no longer to be censorship of the press, and I am not aware of any circumstance where the Soviet government stopped a particular report about the USSR's hardships under Glasnost.

For example, the people in the USSR became very well aware of the Chernobyl Nuclear Disaster (though one could argue that Gorbachev had to do this, as many other countries caught the radiation readings). Equally, newspapers began to report on high mortality rates, poor housing, the decline in industrial growth, the agricultural problems, and any number of things you can think of that went wrong in the USSR over the next few years.

In short, no one was aware that the USSR would collapse, but due to relaxation of censorship, many newspapers began to report on all the growing problems the Soviet Union was facing.