It won't like the zero in
Set XTemp = Range("C1").Resize(x, 0)
because you can't have a selection zero columns wide I think.
Also is x properly defined somewhere? I suspect you mean to use
amtrows there?

The line
ActiveChart.SetSourceData Source:=Sheets("Sheet5").Range(XTemp),
PlotBy _ :=xlColumns

will confuse vba  as you've already defined XTemp as a range so
Range(Xtemp) may be a little dodgy.
Set the range to include sheet info as well:
Set XTemp = Sheets("Sheet5").Range("C1").Resize(amtrows, 1)

and change the SetSource line to:
ActiveChart.SetSourceData Source:=XTemp, PlotBy:=xlColumns

It also helps to be explicit about which sheet Range("A8") is on:
amtrows = Sheets("Sheet5").Range("A8").Value


Macro in full:

Dim amtrows As Long
Dim XTemp As Range
amtrows = Sheets("Sheet5").Range("A8").Value
Set XTemp = Sheets("Sheet5").Range("C1").Resize(amtrows, 1)
ActiveSheet.ChartObjects("Chart 2").Activate
ActiveChart.ChartArea.Select
ActiveChart.SetSourceData Source:=XTemp, PlotBy:=xlColumns
ActiveWindow.Visible = False

Since I posted my code and problem
earlier today, I have modified the code (see below). Despite my
modifications, I am still receiving errors.

I have a VBA coding problem  with the Sub below. All lines work fine
until I reach the lines commented with asterisks * * *. Those 7 lines
of Series Collection code are puzzling me. Could you please go over
my code and tell me (show me) what I'm doing wrong ?

Sub ChartTrial()

Dim DataRange As Range
Dim UpperLmtRange As Range
Dim LowerLmtRange As Range
Dim TimeFrame As Range
Dim DataName As String
Dim UpperLmtName As String
Dim LowerLmtName As String
Dim XMax As Long
Dim XMin As Long
Dim XUnit As Long
Dim amtrows As Long

'assign names
DataName = "Region1"
UpperLmtName = "Upper Limit"
LowerLmtName = "Lower Limit"

'determine axis scale limits
XMax = Range("Sheet3!B6").Value
XMin = Range("Sheet3!B7").Value
XUnit = Range("Sheet3!B8").Value

'get count of data rows to plot
amtrows = Range("Sheet5!C4").Value

'set-up ranges to plot
Set TimeRange = Range("Sheet3!C7").Resize(amtrows, 1)
Set DataRange = Range("Sheet3!D7").Resize(amtrows, 1)
Set UpperLmtRange = Range("Sheet3!E7").Resize(amtrows, 1)
Set LowerLmtRange = Range("Sheet3!F7").Resize(amtrows, 1)

ActiveSheet.ChartObjects("Chart 1").Activate
ActiveChart.ChartArea.Select

'set axis scale values
With ActiveChart.Axes(xlValue)
.MaximumScale = XMax
.MinimumScale = XMin
.MinorUnit = XUnit
End With

'assign data range
ActiveChart.SetSourceData Source:=DataRange, PlotBy:=xlColumns

ActiveChart.SeriesCollection(1).Name = DataName
ActiveChart.SeriesCollection(1).Values = DataRange

'* * *errors occur with the next 7 lines
ActiveChart.SeriesCollection(2).Name = UpperLmtName
ActiveChart.SeriesCollection(2).Values = UpperLmtRange
ActiveChart.SeriesCollection(3).Name = LowerLmtName
ActiveChart.SeriesCollection(3).Values = LowerLmtRange
ActiveChart.SeriesCollection(1).XValues = Range(TimeFrame)
ActiveChart.SeriesCollection(2).XValues = Range(TimeFrame)
ActiveChart.SeriesCollection(3).XValues = Range(TimeFrame)
'* * * errors end here

ActiveWindow.Visible = False

Range("A1").Select

End Sub

1. Remove or comment out this line:
ActiveChart.SetSourceData Source:=DataRange, PlotBy:=xlColumns

Since you already have the chart  in place you don't need to redefine
and plot. Since DataRange is only one column wide you end up only
plotting one series, therefore later when you refer to series 2 and 3
etc. it balks. Just make sure you have 3 series and the chart type as
you like it before running the macro.

2. The line
Set TimeRange = Range("Sheet3!C7").Resize(amtrows, 1)
I believe should be:
Set TimeFrame = Range("Sheet3!C7").Resize(amtrows, 1)
since you have no Dim statement for TimeRange, but have one for
TimeFrame!

3. You've done the same as before with the three lines:
ActiveChart.SeriesCollection(1).XValues = Range(TimeFrame)
ActiveChart.SeriesCollection(2).XValues = Range(TimeFrame)
ActiveChart.SeriesCollection(3).XValues = Range(TimeFrame)

TimeFrame is already a range so amend to:
ActiveChart.SeriesCollection(1).XValues = TimeFrame
ActiveChart.SeriesCollection(2).XValues = TimeFrame
ActiveChart.SeriesCollection(3).XValues = TimeFrame