Hi Nat..what size is the pot?
The leaves that haven't any brown look most colorful, but D. Lemon-Limes prefer a bit more light than most Dracaenas. Lack of sufficient light is one reason it's (a bit) spindly.
How much fertilizer do you use? Dracaenas don't need much fertilzier since they're slow-growing. Especially if you're using chemical fertlizers and/or the plant isn't getting much sun. Overfertilizing can cause leaves to brown.
If you feel the pot is too large, (more than 1-2" surface between rootball and inner pot) then underpot. I think your pot is clay which helps soil dry out faster. My Lemon-Lime soil dries out more than 2" from the top before given a drink. Soil should feel crumbly between waterings.

It's a good idea keeping water in a container at least 24 hours before watering. I save old milk containers filled with water, then when plants are thirsty, just pour..Refil milk container for future use.

If it was my Lemon-Lime, I'd remove all brown leaves. Any brown edges should be clipped, but leave about 1/8" brown on plant, otherwise it might spread.
Unpot and check roots..if pot is too big, plant would go in a smaller, plastic container.
Although Peat works for acidic loving plants, instead of peat I'd use All Purpose Soil and Perlite..
Every leaf, top and bottom, would be inspected for insects. And we can't forget to check soil.
After repotting, plant would go in the sink/shower and get hosed. Misting helps with humidity and keeps foliage free of dust. If you're running A/C this time of year, the air is dry. If you're not into misting then hose in sink/shower once a week.
Last, place Lemon-Lime in a brighter area..It doesn't need direct south or west summer sun, (west or south is perfect in winter) but a little direct, plus bright light keeps colors pronounced. Good luck, Toni

Unfortunately, this is the normal growth habit of this plant. Eventually the lower leaves fall off, and it looks spindly. You can try to cut off and reroot the top, buy a new one, or just live with it as is.

You can repot if you want to, I would suggest a soilless potting mix like ProMix or Fafards, maybe throw in some extra perlite if have some. The yellow brown leaf suggests a bit too much fertilizer and/or water. You can probably cut back a bit on both. You can increase the light a bit too if you want to.

But basically your plant looks normal for a 5 year old plant.

Watergal, when you say, the normal growth of this plant, do you mean the Lemon-Lime specifically or are you including Dracaenas like Janet Craig, too? Toni

Toni, all dracaenas eventually drop the lower leaves and don't regrow them. So the objective is to get the watering, soil and light just right to minimize the leaf drop. They will still lose leaves over time, but less-than-ideal culture makes them drop sooner.

All dracaenas, including lemon-lime and the other warneckiis do this. So do Janet Craig and marginata (dragon tree) and massangeana (corn plant) and any others you may run across.

My Lemon Lime is doing great. It is one tall stalk with luscious curving leaves. How may I encourage it to grow branches or does that not happen with this plant?

My Lemon Lime is doing great. It is one tall stalk with luscious curving leaves. How may I encourage it to grow branches or does that not happen with this plant?

I love this plant. I've also had fun getting the other varieties of dracaena that is being offered in the stores.

Mine was bought as an 8 inch plant, potted up to 10 and has taken off. It grows in front of a large office window.
I do mist and fertilize

Susana111, if you cut off the top, there's a high probability it will regrow multiple tops, but no guarantees.

Cut off the top, and put it back in the pot (bury the stem). It should reroot. The trunk that was cut should also sprout again from near the top and continue to grow.