Here's the best advice I've ever heard on this subject... I wish I had heard and understood this about 25 years ago when I was first starting out.
Playing any instrument is a physical act. When you're playing bass, especially as a beginner, you tend to focus on the physical part of it - i.e., getting your fingers to hit the right fret, moving them to the next fret, plucking the right string, etc. When you are unable to play a difficult passage, like a quick run of notes at the end of a verse, you have a natural tendency to think of it as a physical limitation - like you are unable to make your fingers do what they need to do. Here's the problem: it's not a physical limitation; if you have been playing for at least a few months and you are able to play in a cover band, then you have developed probably 90% or more of the physical ability (muscle strength, dexterity, etc) that you need to play pretty much any of those parts.
The real thing that you are developing as you improve as a player is the mental link between your brain and your fingers. When your brain decides to play a note, it sends a signal to your hands to tell them what to do. What separates a novice player from an accomplished one is not necessarily that the accomplished player has more dexterity, strength, etc., it's because they know music and their instrument better, and because the signal from their brain to their hands is so loud and strong. I think that's the real reason why some people can pick up an instrument so much faster than others - because their brains naturally produce strong, loud signals.
So what can you do with this concept? When you are working on one of those difficult parts, don't focus on your hand - in other words, don't concentrate on getting your fingers to do what you want them to do. Instead, work on the part slowly until you know all the notes and can play it at half-speed. Then close your eyes, take a deep breath, and then "hear" the part in your head as loud as you can - mentally "shout out" the notes as loud as possible. Right after that, open your eyes and try to play it. What you are doing is you're creating a strong signal that your brain is sending to your fingers - the idea is to make the signal so loud that your fingers have no choice but to play it. Work to make this a habit - focus on hearing the song in your head rather than making your fingers to the work. I think you will be surprised at how much this can improve your playing.
I know this is all counter-intuitive and it doesn't make much sense. Think about it and work on it, and once you get it, it will help. In the meantime,
watch this YouTube video. Be patient and listen to it all; especially pay attention to the woman at the end who plays a short passage on the piano.
Also listen to what everyone else has said in this thread. I'm pretty sure we've all been through what you're experiencing right now. The bottom line is that it all takes time and practice - don't be too hard on yourself, and stick with it.
-Y