Continuity mistake: Near the beginning when Colleen and Becky are talking to the boy, there is a boy walking in the background. When he loses his hat, he is behind the boy talking to the girls. In the next shot he is already behind the two girls, which is quite a great distance to cover in a split-second.
Continuity mistake: Jesse and Brian enter the President's office from the secret passage shortly after Sen. Steward and Capt. Burgoyne leave. When the two boys hear the door rattle, Jesse whispers to Brian that somebody's coming and they race off Right. The President's desk is still in 3/4 view, and remains so as Burgoyne re-enters and goes to the desk, noticing the open cabinet door, which he checks. He hears movement and voices a threat, then goes to check the curtains only to find nothing. The camera then pans down to show Jesse and Brian hiding scared under the desk. Given the brief timespan, the two boys could not possibly have moved there without Burgyone noticing their presence.
Other mistake: In the scene where Dorothy decides to join the church picnic (just before her operation) Grace is talking to her and hands her a glass of cider as Robert E. approaches. The sky is partly cloudy, but the lighting changes slightly during the shot, and at one moment, appears too bright on Dorothy just before she sets the glass down and walks away.
Audio problem: Just before Dorothy has her operation, Mike is looking through her medical books when Sully pays a visit of comfort. During their tender moment, you can hear slight foot shuffling from the camera crew in the foreground - which is not coming from the outside. Also, you hear a slight grunting sound during a slight pause in their conversation.
Continuity mistake: When the whole town comes to support Matthew, we can see Hank and the reverend walking up behind him. A few seconds later, we see them both again, now they're walking with Colleen and Brian, and they have just come around the corner to see what all the fuss is about.
Plot hole: Marjorie's tragic death: Though Colorado Springs is a considerable distance from the Atlantic coast, the Quinns should have honored Marjorie's death with a town vigil. They should also have had her body properly transported back home to Boston, to be buried alongside her Quinn relations, as was (and still is) common respect for soldiers, family members, and dignitaries - instead of burying her in the town cemetery, where she did not belong.
Visible crew/equipment: Loren wakes Dorothy in the middle of the night to mention selling the store and taking her on a trip to New York. While seated at her bedside, you can see shadows of a crew member on the bed pillows next to her, and hear some muffled foot movement while Loren is seated, talking to her.
Plot hole: In the ending scene, after Mike's serious fall, she has a probable concussion and Sully desperately keeps her awake to avoid her slipping into a coma. They talk all night, reminiscing. During this, Mike experiences periodic pain flashes from her fall. They conclude their evening by discussing having another child - then immediately share a rather intense intimate moment. This would be impossible. Even if pain and swelling goes down, one does not just jump back in the game again, and win that 500 yard dash. Realistically, Mike and Sully would have returned home to have Andrew examine her thoroughly, before deeming her ready to be intimate again.
Factual error: Pilot: When the woman is giving birth and Dr. Quinn has to perform a c-section, she gives the baby to Mrs. Cooper, and after about a minute, she gives the baby to the mother. Dr. Quinn then leans towards Mrs. Cooper and they smile happily; she doesn't attempt to continue the "surgery". Even nowadays doctors need several minutes to close up again after a c-section, so there is no way that Dr. Quinn was anywhere near finished. Leaving the patient cut open on the bed is pretty irresponsible, something she would never do as she has to prove that she is a good M.D.