I have the following element:

<!-- language: lang-html -->
<div class="bottom-est">Bottom-Est</div>
<!-- language: lang-css -->
.bottom-est {
  position: fixed;
  width:100%;
  bottom: 0;
  border: 1px solid black;
  background-color: orange;
}

Can I place another fixed position element right above it?

If it has a fixed/known height, I can just add bottom:##px to displace it:

<!-- language: lang-html -->
<div class="bottom-ish">Bottom-Ish</div>
<!-- language: lang-css -->
.bottom-ish {
  position: fixed;
  width:100%;
  bottom: 18px;
  border: 1px solid black;
  background-color: yellow;
}

Which will accomplish the resulting goal:

Screenshot of Goal

But is there a way to do this that doesn't require knowing the height ahead of time. I can't modify .bottom-est at all, but I can modify .bottom-ish as much as I want.

<!-- begin snippet: js hide: true console: true babel: false --> <!-- language: lang-css -->
body {
  margin:0;
  padding:0;
}

.bottom-ish {
  position: fixed;
  width:100%;
  bottom: 18px;
  border: 1px solid black;
  background-color: yellow;
}
.bottom-est {
  position: fixed;
  width:100%;
  bottom: 0;
  border: 1px solid black;
  background-color: orange;
}
<!-- language: lang-html -->
<div class="bottom-ish">Bottom-Ish</div>
<div class="bottom-est">Bottom-Est</div>
<!-- end snippet -->

With CSS alone, the answer is NO.

Once you apply position: fixed to an element you remove it from the document flow. This means that other elements don't respect the fixed element's space. In fact, they treat the fixed element as though it doesn't even exist.

So unless you know the height of the fixed element, you can't position another element to stack right above it.