Simplistic Lava Lamp

Python 2, 117 113 108 107 106 105 bytes

from random import*
def f(n):a=['']*5;exec"i=randint(0,(n>44)+(n>59)<<1);a[i]=(a[i]or 0)+1;"*1000;print a

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Returns a reversed list (bottom first)

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Version inspired by the stackoverflow answer in the comments (edgecases are more likely):

Python 2, 129 bytes

from random import*
def f(n):a=sorted([1000]*5+sample(range(1001)*5,(n>44)+(n>59)<<1));print[y-x or''for x,y in zip([0]+a,a)[:5]]

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JavaScript (ES6), 78 bytes

Returns a reversed array where empty levels are filled with a space.

t=>(a=[...'     '],g=k=>k?g(k-1,a[Math.random()*(t>59?5:t<45||3)|0]++):a)(1e3)

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Commented

t => (                      // t = input
  a = [...'     '],         // a[] = output array, initially filled with 5 spaces
  g = k =>                  // g = recursive function taking an iteration counter k
    k ?                     //   if k is not equal to zero:
      g(                    //     do a recursive call:
        k - 1,              //       decrement k
        a[                  //       update a[]:
          Math.random() * ( //         pick a random slot:
            t > 59 ? 5 :    //           among all 5 slots if t > 59
            t < 45          //           force the 1st slot if t < 45
            || 3            //           among the 3 first slots otherwise
          ) | 0             //         round the above result to an integer
        ]++                 //       increment the wax amount on this slot
      )                     //     end of recursive call
    :                       //   else:
      a                     //     stop recursion and return a[]
)(1e3)                      // initial call to g() with k = 1000

R, 85 84 bytes

function(n)write(ifelse(t<-table(cut(runif(1e3,2*(n<60)+3*(n<45),5),0:5)),t,""),1,1)

-1 byte thanks to @Giuseppe

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Explanation (ungolfed):

function(n){
      # Generate 1000 random uniform numbers in [5,5] (if n<45),
      # in [2,5] (if 45<=n<60) and in [0,5] (if n>=60).
    x = runif(1e3,2*(n<60)+3*(n<45),5) 
      # Code each by the number of the interval it falls in (0,1],(1,2]...(4,5]
    cx = cut(x,0:5)
      # Tabulate the intervals. Because cut() returns a factor,
      # zero counts are included 
    t = table(cx)
      # Vector-wise replace zero elements with "" and cat out, 1 per line.
    t1 = ifelse(t,t,"")
    write(t1,1,1)
}