You can simply use a check constraint however I would personally restructure my schema. Essentially what you seem to be doing is serializing a lot of data into a path, and then you want to be sure it's correct. That's kind of yuck, from my perspective.

CREATE TABLE foo (
  id    serial,
  path  text
    CHECK ( path ~ '/files/client/\d+/(attachment|user)/\d+/(main|thumbnail)' )
);
INSERT INTO foo(path) VALUES 
  ('/files/client/743052/attachment/2123598/main'),
  ('/files/client/743052/attachment/2123598/thumbnail'),
  ('/files/client/1475296/user/3541234/main'),
  ('/files/client/1475296/user/3541234/thumbnail');

INSERT INTO foo(path) VALUES ('/STUD/EVAN/beefcake_hotstuff.jpg');
ERROR:  new row for relation "foo" violates check constraint "foo_path_check"
DETAIL:  Failing row contains (6, /STUD/EVAN/beefcake_hotstuff.jpg).

I would probably go the opposite direction and normalize to some degree.

Normalizing

Normalizing would look something like this,

CREATE SCHEMA aws;

CREATE TABLE aws.client (
  client_id serial PRIMARY KEY
);
CREATE TABLE aws.attachment (
  client_id       int REFERENCES client,
  attachment_id   int PRIMARY KEY,
  name            text
);
CREATE TABLE aws.user (
  client_id       int REFERENCES client,
  user_id         int PRIMARY KEY,
  name            text
);

SELECT '/files/client' || client_id || '/attachment' || attachment_id || '/' || a.name
FROM aws.attachment AS a
JOIN aws.client AS c USING (client_id);