Last week, an anguished reader phoned and told me she had been trying to reach Social Security for several days but the line was always busy. She wanted to know whether service had already been cut back.

I guessed that the line was probably out of order and that I knew what to do about it because I had just had a similar experience in trying to reach the Metropolitan Police Department's Sixth District to check out an accident report.

Three times I had dialed the number listed in the phone book and let it ring 20 times, but nobody answered.

Not until I phoned a communications supervisor did I learn that the station's telephone equipment was malfunctioning. Sometimes the 6-D phone rang and was answered. But at other times it failed to ring. Ma Bell's repairmen got on the problem at once.

However, the number for Social Security information here (953-3600) rang busy constantly for me, as it had for the reader, so I called my friend Jim Brown at Social Security and told him about it. Three days later, I received a letter from Commissioner of Social Security John A. Syahn, who is headquartered in Baltimore. Syahn said he was sorry to hear that callers were having trouble getting through, particularly so because in recent years Social Security has been trying to make it easier for people to phone in. He added:

"Special teleservice centers in large metropolitan ares have been established to handle these calls.

"In the Washington area, for example, most calls about Social Security are handled by our teleservice center in Laurel. The people there will either answer questions from the caller or refer the caller to a district or branch office for further assistance.

"The Laurel teleservice center, as well as other centers throughout the country, are getting additional personnel as well as new telephone equipment to handle the increasing volume of telephone calls resulting from new programs and increasing public awareness. Improving beneficiary service is one of the major items on my agenda and I intend to take whatever steps are necessary in seeing that the public is served well, promptly and efficiently."

I hope Commissioner Syahn is given an adequate budget and will be able to achieve his goal. Cuts in service could cause millions of Americans who now depend upon Social Security to look back on 1981 as "the good old days."

Meanwhile, here's a tip from an insider at Social Security:

Try to avoid calling teleservice centers during the first week of each month, because that's when their lines are busiest.

I made a test call to 953-3600 yesterday and got through to the switchboard on the first try -- no busy signal. A recorded announcement said all the information clerks were busy but I would be answered in turn. And after about a 10-minute wait my call was answered. IT HAPPENS

My call to 6-D headquarters followed a report from David Watkins that he had almost been run off the road by a speeding driver on Rte. I-295 a few minutes before.

Watkins said the speeder had caused several accidents and would have quickly disappeared from sight except that "all four tires flew off his car. He abandoned it and ran away."

Watkins wrote down the license number and tried to report the matter to the police but encountered a dispatcher who wanted to know at what address all this had taken place. Watkins said, "On Kenilworth Avenue, between Benning and Eastern," but that wasn't good enough for the dispatcher, who insisted that a specific address is mandatory because "that's the way the computer is set up."

When the phone bells finally rang in 6-D, I was able to ascertain that the dispatcher had been misinformed. The police computer understands locations identified by cross streets as well as by specific addresses.

I consider the lapse minor because, despite the high pressure under which they work, police dispatchers seldom make mistakes. They're really good at their jobs, and every human being is entitled to an occasional goof.

Incidentally, when I asked whether the police had been able to find out why the speeder had been driving at such a pace, the policeman at the other end said, quite sadly, "Have you ever been on 295? Half the people on 295 drive that way."