Yes, it was relatively simple to fake identities.
A famous, famous example of this is shown in Day of the Jackal, where Frederick Forsyth included a character who faked his identity using a very simplistic method that had been known about for decades and took until the noughties to actually rectify.
Basically, he showed that someone could trawl a village or location for graves. The person looking would search for someone who had died young, but if they'd been alive, would be roughly the same age as the person looking.
Next, the person would go to the local registry office and buy the birth certificate of the deceased (no identification was needed for this).
Finally, they would apply for a passport (the only piece of evidence needed to do this was a birth certificate).
Doing this, it was utterly simplistic to fake someone's identity.
Now, Forsyth's example related specifically to the UK. However, in the US, the law wasn't all that different.
It used to be the case that US citizens didn't get social security numbers until they got their first job.
Therefore, someone looking to assume a fake identity could again search for a deceased person who, if alive, would be roughly their age. They could again purchase their birth certificate. Finally, they could apply for a social security number and explain their reasons for doing so (let's say the person stealing the identity was 30 - they could explain they didn't have a job for ten years for any reason they liked).
Another factor to consider was that the punishment for faking identities used to be far less severe than it is now. For criminals with a string of offences on their record, the relatively minor crime of faking their identity was worth it - whereas nowadays, it would carry a much more severe sentence.
So in summary - it was very simple to fake identities until the digitalisation of the data (and even now, it's still far from foolproof).