Continuity mistake: After Eugene takes the stage to face Jack Butler, everyone (including Eugene) turns when Scratch walks into the room. When Scratch sits down and Eugene turns back to face Jack Butler, Eugene's hat is gone.
Other mistake: When Eugene discovers that it really is Blind Dog Fulton, he (Fulton) says that he needs to get out to help Eugene find that lost song. Eugene says that he will be here at "5:00 a.m." When they breakout - which took all of about maybe 10 minutes, it's very light for the morning time. Looks to be maybe 6:30 by daylight savings time.
Suggested correction: Aside from the fact that daylight continuity is seldom strictly observed in any film, you should be aware that dawn on the East Coast visibly begins breaking over the Atlantic at around 4:30 to 5:00 AM during Daylight Savings Time. The sun may not have cleared the horizon, but the sky is really quite bright in New York City (and the whole East Coast) by 5:00 AM.
The time of sunrise varies and, although it isn't pointed out what month they left the nursing home, it is suggested that it was during the academic year - not a summer month when sunrise would be within the earliest time frame. Astronomical, nautical and civil twilight (all 6° or more below the horizon) do not radiate the amount of light shown in the movie. Moreover, it did not appear to get lighter out as they traveled, suggesting they departed after daylight, which would be well after 5 AM.
Continuity mistake: The amount of whiskey increases in Eugene's free shot at the bar.
Continuity mistake: Lloyd shut the bathroom door when he went in to start the shower (to make the water hot for Frances). When Eugene went to the motel room to get Frances out, a shot of the bathroom door showed it ajar (a few inches) with steam coming out. The door was closed when Lloyd came out of the bathroom. (00:44:20)
Continuity mistake: Eugene started the guitar competition wearing the "slide" (on his left hand), which Willie Brown previously told him he had to use if he wanted to play Blues. At one point during the competition, Eugene did not have the slide on but he was wearing it after he won. (01:27:10 - 01:31:42)
Continuity mistake: The truck drove around the bend, past the Highway 61 sign, and stopped near the mailbox. After Eugene [Ralph Macchio] and Willie Brown got off the back of the truck, they were standing at/near the bend in the road, almost directly across from the North 61 highway sign. The bend in the road also looked much sharper. (00:30:05 - 00:30:42)